The area of triangle PQR in this case is 14.924
In geometry, the concept of vectors is very useful when dealing with points, lines, and planes in three-dimensional space. Vectors have both magnitude and direction, and they can be added, subtracted, and multiplied by scalars.
To find a nonzero vector orthogonal to the plane through the points P, Q, and R, we can use the cross product of two vectors in the plane. A cross product of two vectors gives us a vector that is perpendicular to both of them, so it will be orthogonal to the plane as well.
Let's take the vectors from P to Q and from P to R as our two vectors:
u = Q - P = <5-3, 2-1, 4-1> = <2, 1, 3>
v = R - P = <8-3, 5-1, 3-1> = <5, 4, 2>
To calculate the cross product u x v, we can use the following formula:
u x v = <(1)(2) - (3)(4), (3)(5) - (1)(2), (1)(4) - (2)(5)> = <-10, 13, -2>
So the vector <-10, 13, -2> is orthogonal to the plane through P, Q, and R. Note that this vector is nonzero because at least one of its components is not zero.
To find the area of triangle PQR, we can use the fact that the area of a triangle is half the magnitude of the cross product of two of its sides. Let's take the sides from P to Q and from P to R again, and calculate their cross product:
u x v = <(3)(1) - (-6)(3), (-6)(2) - (3)(3), (3)(3) - (3)(2)> = <21, -21, 3>
So the vector <21, -21, 3> is orthogonal to the plane through P, Q, and R. Again, we can use the cross product of u and v to find the area of triangle PQR, which is:
|u x v| = sqrt(21^2 + (-21)^2 + 3^2) = √(891)
|u x v| = 1/2 * √(891) =14.924
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If f(x) = ln(x), what is the transformation that occurs if g(x) = ln(x + 2)
The transformation from f(x) to g(x) is a horizontal shift to the right by 2 units to obtain the graph of g(x).
What do you mean by Transformation?In mathematics, transformation refers to a change in position, shape, size, or orientation of a figure or a function. There are various types of transformations, including translation, rotation, reflection, dilation, and more.
A translation is a transformation in which a figure is moved to a new position on the coordinate plane, while keeping its original size and orientation intact. A rotation is a transformation in which a figure is turned about a fixed point, called the center of rotation. A reflection is a transformation in which a figure is flipped over a line, called the line of reflection. A dilation is a transformation in which a figure is enlarged or reduced, while keeping its shape intact.
Transformations are used in various areas of mathematics, including geometry, engineering, computer graphics, and more. They provide a way to model real-world objects and processes and to solve problems related to size, position, and orientation. Understanding transformations is a fundamental aspect of mathematical skills and is crucial in many areas of study and research.
The transformation from f(x) = ln(x) to g(x) = ln(x + 2) is a horizontal shift to the right by 2 units. This is because when you replace x with x + 2 in the logarithmic function, you are shifting the graph to the right by 2 units. The x-intercepts of the graph of f(x) will be at x = 1 (since ln(1) = 0), whereas the x-intercepts of the graph of g(x) will be at x = -2 (since ln(2) = 0). So, in essence, the transformation is simply a horizontal shift of the graph of f(x) to the right by 2 units to obtain the graph of g(x).
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Tamika leans a 30-foot ladder against a wall so that it forms an angle of 65∘ with the ground. How high up the wall does the ladder reach? Round your answer to the nearest hundredth of a foot if necessary.
Answer:
64.34 feet.
Step-by-step explanation:
To solve this problem, we can use trigonometric ratios, specifically the tangent ratio.
Let's denote the height up the wall that the ladder reaches as "h". We can then set up the following equation:
tan(65°) = h/30
To solve for h, we can multiply both sides by 30:
30 tan(65°) = h
Using a calculator, we can find that tan(65°) is approximately 2.1445. Multiplying this by 30 gives us:
h ≈ 64.34 feet
Therefore, the ladder reaches approximately 64.34 feet up the wall. Rounded to the nearest hundredth of a foot, the answer is 64.34 feet.
On Monday, he made 7 cakes. On Lionel makes cakes for the bake sale.
On Monday, he made 7 cakes. On Tuesday, he works an additional 2 hours and has a total of 13 cakes.
Determine how many cakes he makes in one hour. Tuesday, he works an additional 2 hours and has a total of 13 cakes.
Determine how many cakes he makes in one hour.
To determine how many cakes Lionel makes in one hour, we can use the following formula:
cakes per hour = total cakes / total hours worked
On Monday, Lionel made 7 cakes in some number of hours, so we don't know how many cakes he made per hour that day. However, on Tuesday, he made an additional 6 cakes (13 total cakes - 7 cakes made on Monday), and he worked for 2 hours longer than he did on Monday. Therefore:
cakes per hour on Tuesday = 6 cakes / 2 hours = 3 cakes per hour
So Lionel made 3 cakes per hour on Tuesday. We don't have enough information to determine how many cakes he made per hour on Monday.
please help me !!!!!
Answer:28
Step-by-step explanation: because add z and x
A box was made in the form of a cube. If a second cubical box has inside dimensions four times those of the first box, how many times as much does it contain?
(A) 4
(B) 8
(C) 16
(D) 64
(E) none of these
Answer:
Since the second box has inside dimensions four times those of the first box, its volume is $(4a)^3 = 64a^3$ times as much as the volume of the first box, whose volume is $a^3$. Therefore, the answer is (D) 64.
Complete the sentences to explain how to find the dimensions of the square.
The absolute value of the x-values of the coordinates of A and B = |2 - 5| = 3 units.
The absolute value of the y-values of the coordinates of B and C = |6 - 3| = 3 units.
How to Find the Dimension of a Square on a Coordinate Plane?If two points lie on a coordinate plane, the distance between them represents the length of the segment both points form, which is determined by finding the absolute difference between the values of their coordinates that lie on the same axis.
Given that a square has the points, A (2, 6), B (5, 6), C (5, 3), D (2, 3), its dimensions can be found as explained below:
Length of AB is found by calculating the absolute value of the x-values of the coordinates of A and B, while to find the length of BC, is the absolute value of the y-values of the coordinates of B and C.
Therefore:
AB = |2 - 5| = 3 units
BC = |6 - 3| = 3 units
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find the area a of the sector of a circle of radius 100 meters formed by the central angle 1/8 radian
The area of a sector of a circle with radius 100 meters and central angle 1/8 radian is approximately 625 * π square meters.
The area of a sector of a circle with radius 100 meters and central angle 1/8 radian can be found using the formula:
A = [tex](\theta/2\pi )*\pi r^{2}[/tex]
where A is the area of the sector, Θ is the central angle in radians, π is Pi (approximately 3.14), and r is the radius of the circle.
Plugging in the given values, we have:
A = [tex](\frac{\frac{1}{8}}{{2\pi }} )*\pi *100^{2}[/tex]
A =[tex](1/16)*\pi *100^{2}[/tex]
A = [tex](\pi /16)*100^{2}[/tex]
A = (π/16) * 10000
A = 625 * π square meters
So, the area of the sector of the circle with radius 100 meters and central angle 1/8 radian is approximately 625 * π square meters.
The area of a sector of a circle is a portion of the circle's area enclosed by two radii and an arc. The central angle of the sector determines the fraction of the circle's circumference that the arc represents, and therefore the fraction of the circle's area that is enclosed by the sector.
To find the area of a sector, we first need to find the central angle in radians. The central angle is the angle formed by two radii at the center of the circle that intercept the circumference of the circle at the endpoints of the arc.
The formula for the area of a sector is given by:
A = [tex](\theta/2\pi )*\pi r^{2}[/tex]
where A is the area of the sector, Θ is the central angle in radians, π is Pi (approximately 3.14), and r is the radius of the circle.
In the given problem, the radius of the circle is 100 meters and the central angle is 1/8 radian. We plug these values into the formula and simplify to get the final answer.
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6. Tyler went to the gym on 22 of the last 31 days. Which of the following is closest to the
percent of days that Tyler went to the gym?
(1) 66%
(3) 70%
(2) 67%
(4) 71%
Answer:
71%
Step-by-step explanation:
We know
Tyler went to the gym on 22 of the last 31 days.
Which of the following is closest to the percentage of days that Tyler went to the gym?
We take
22 divided by 31, times 100 = 71%
So, the answer is 71%
Let X1, X2, X3,.......Xn denote a random sample from the uniform distribution on the interval a:X-1/2 and b:Xn-n/n+1 find the efficiency of a and b
The efficiency of the a and b from the uniform distribution on the interval a:X-1/2 and b:Xn-n/n+1 is x.
The probability distribution known as the uniform distribution comes in two varieties: the discrete case and the continuous case. Both types of uniform distributions, as their names imply, have constant or uniform probabilities for all possible values of the random variable.
The random sample X1,X2,....,Xn is taken from the uniform distribution on the interval (0,θ).
δlog(L(x,θ))/δθ = -n/θ + ∑(xt)/θ² = 0
n/θ = ∑(xt) /θ²
θ = x
Now derivate it again with respect to θ,
∂²log(L(x;θ)) / ∂θ²
=nθ²−2∑t(xt) / θ³<0
Since, the second derivative is less than 0, ^θ is maximum likelihood estimate of θ.
Therefore, the maximum likelihood estimate of θ is ¯x.
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Respond to the following discussion prompt.
Jason uses his credit card for all his monthly expenses. He pays only the minimum payment required every month. The credit card company charges him interest that is compounded monthly and a penalty for a missed payment. Based on how principal and interest compound over time, is he doing the right thing by making the minimum payment? Justify your answer.
it is not advisable to make only the minimum payment on a credit card if Jason wants to minimize the amount of interest he has to pay and pay off his debt faster.
What is Credit card?A credit card is a payment card issued to users to enable the cardholder to pay a merchant for goods and services based on the cardholder's accrued debt.
Jason may not be doing the right thing by making only the minimum payment on his credit card, depending on the interest rate and penalty charges.
When a credit card holder makes only the minimum payment, the remaining balance accrues interest, which is added to the principal amount.
This means that the total amount owed on the credit card increases every month, even if no additional purchases are made.
The interest rate charged by the credit card company can significantly impact how much Jason ends up paying over time.
Hence, it is not advisable to make only the minimum payment on a credit card if Jason wants to minimize the amount of interest he has to pay and pay off his debt faster.
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MTH 115 Lesson 8 Practice Problems
Answer Key
1. A credit card charges 23% APR and a minimum of 2.5%. If you pay the minimum on a balance of $450 what would the decrease in your balance be? Round to the nearest cent $2.63.
2. A month has 30 days in it. You post a payment of $600 on the 15th of the month. The Balance was $2000 before the payment. What is the interest on the average daily balance if the APR is 23%? $32.58
3. The monthly payment on a house loan is $876. The purchase price of the house was $125,000. If you pay this for 30 years how much interest did you pay? $190,360
4. A house costs $275,000. The rate is 7.3% with monthly payments for 30 years. What is the payment? Round to the nearest dollar. $1885
A is the answer
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Hint(s) Check My Work A 2018 Pew Research Center survey found that more Americans believe they could give up their televisions than could give up their cell phones (Pew Research website). Assume that the following table represents the joint probabilities of Americans who could give up their television of cell phone. Excel File: data04-37.xlsx Could Give Up Television Yes No Could Give Up Yes 0.31 0.17 0.48 Cellphone No 0.38 0.14 0.52 0.69 0.31 a. What is the probability that a person could give up her cell phone (to 2 decimals)? that b. What is the probability that a person who could give up her cell phone could also give up television (to 2 decimals)? c. What is the probability that a person who could not give up her ell phone could give up television (to 2 decimals)? d. Is the probability a person could give up television higher if the person could not give up a cell phone or if the person could give up a cell phone? The probably a person could give up television if they could not give up a celone is Select your answer than the probability a person could give up television if they could give up a cellphone
a. The probability that a person could give up her cell phone would be 0.69.
b. Probability that a person who could give up her cell phone could also give up television would be 0.45
c. The probability that a person who could not give up her ell phone could give up television would be 0.55.
d. Probability of giving up television is higher for people who could not give up their cell phone than for those who could give up their cell phone.
What is probability?
Probability is an area of mathematics that deals with numerical descriptions of how probable an event is to occur or how likely a statement is to be true. The probability of an event is a number between 0 and 1, where 0 denotes the event's impossibility and 1 represents certainty.
a. The probability that a person could give up their cell phone is the sum of the probabilities in the "Could Give Up" column: 0.31 + 0.38 = 0.69.
Therefore, The probability that a person could give up her cell phone would be 0.69.
b. The probability that a person who could give up their cell phone could also give up television is the conditional probability of "Could Give Up Television" given "Could Give Up Cellphone." This is calculated by dividing the joint probability of "Could Give Up Television" and "Could Give Up Cellphone" (0.31) by the probability of "Could Give Up Cellphone" (0.69): 0.31 / 0.69 = 0.4493 or approximately 0.45.
c. The probability that a person who could not give up their cell phone could give up television is the conditional probability of "Could Give Up Television" given "Could Not Give Up Cellphone." This is calculated by dividing the joint probability of "Could Give Up Television" and "Could Not Give Up Cellphone" (0.17) by the probability of "Could Not Give Up Cellphone" (0.31): 0.17 / 0.31 = 0.5484 or approximately 0.55.
d. To compare the probabilities of giving up television for people who could or could not give up their cell phone, we need to calculate the conditional probabilities of "Could Give Up Television" given "Could Give Up Cellphone" and "Could Give Up Television" given "Could Not Give Up Cellphone" respectively.
Probability of giving up television if one could give up a cell phone: 0.31 / 0.69 = 0.4493 or approximately 0.45.
Probability of giving up television if one could not give up a cell phone: 0.17 / 0.31 = 0.5484 or approximately 0.55.
The probability of giving up television is higher for people who could not give up their cell phone than for those who could give up their cell phone.
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Prove that 8 sin26. sin34. sin60. sin86 = root3 sin78
The proof of 8 sin26. sin34. sin60. sin86 = root3 sin78 is given below.
What are the trigonometric identities?Equations using trigonometric functions that hold true for all possible values of the variables are known as trigonometric identities.
Given:
An equation,
8 sin26. sin34. sin60. sin86 = root3 sin78
LHS = 8 sin26. sin34. sin60. sin86
= 0.9085192402
≈ 0.9 to the nearest tenth.
And RHS = 3 sin78
= 0.89023679976
≈ 0.9 to the nearest tenth.
Therefore, 8 sin26. sin34. sin60. sin86 = root3 sin78.
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Triangle EFG is an isosceles triangle.
The figure is not drawn to scale.
What is the measure, in degrees, of angle G ?
Measure of angle G is 62.5°.
What is the sum of all the exterior angles of a regular polygon?For a regular polygon of any number of sides, the sum of its exterior angle is 360° (full angle).
In triangle EFG,
Measure of angle E = 55°
Measure of angle F should be less than 180 = x/2°
By the triangle sum theorem, the Sum of measures of all interior angles of triangle = 180°
m(∠E) + m(∠F) + m(∠G) = 180°
55° + x/2° + x/2= 180°
x= 180° - 55°
x = 125
Measure of angle G = 125/2 = 62.5
Therefore, measure of angle G is 62.5°.
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If the area of square 2 is 64 units^2 and the area of square 3 is 36 units^2,find the area and the side length of square 1.
area of square 1 is 28 square units and its side length is[tex]$2\sqrt{7}$[/tex] units.
How to find the area and length of square?Let the side length of square 1 be x units. Then the areas of squares 1, 2, and 3 are [tex]$x^2$[/tex], 64, and 36 square units, respectively.
Since square 2 has an area of $64$ square units, its side length is [tex]$\sqrt{64}=8$[/tex] units. Similarly, square 3 has a side length of[tex]$\sqrt{36}=6$[/tex]units.
We can now set up an equation based on the fact that the sum of the areas of squares 1 and 3 is equal to the area of square 2:
[tex]$$x^2+36=64$$[/tex]
Solving for x, we get:
[tex]$$x^2=28$$[/tex]
Therefore, the area of square 1 is [tex]$x^2=28$[/tex] square units, and its side length is [tex]$\sqrt{28}=2\sqrt{7}$[/tex] units.
In summary, the area of square 1 is 28 square units and its side length is[tex]$2\sqrt{7}$[/tex] units.
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A scientist is studying wildlife. She estimates the population of bats in her state to be 345,000. She predicts the population to grow at an average annual rate of 1.2%. Using the scientist’s prediction, create an equation that models the population of bats, y, after x years.
y=345,000(0.012)^x
y=345,000(0.988)^x
y=345,000(1.012)^x
y=345,000(1.2)^x
Answer:
Step-by-step explanation:
The equation that models the population of bats, y, after x years can be found using the formula:
y = P(1 + r)^x
where P is the initial population, r is the annual growth rate as a decimal, and x is the number of years.
In this case, the initial population P is 345,000, the annual growth rate r is 1.2% or 0.012 as a decimal, and x is the number of years. Substituting these values into the formula, we get:
y = 345,000(1 + 0.012)^x
Simplifying the expression in the parentheses, we get:
y = 345,000(1.012)^x
Therefore, the equation that models the population of bats, y, after x years is:
y = 345,000(1.012)^x
Step-by-step explanation:
the other answer is correct.
I just want to point out some mechanism used here :
the increase by 1.2%
an increase by x% means adding x% to the original 100%.
so, we end up with (100 + x)%.
"%" just stands for 1/100.
so, if y = 100%, an increase by x% is
y×(100 + x)/100 = y×(1 + x/100)
in our case here that is
345,000 × (1 + 0.012) = 345,000 × 1.012
and since the increase rate of 1.2% applies per year, we multiply the end population of every year by 1.012 for the following year.
so, each year a new factor of 1.012 is being integrated into the calculation.
it goes then
345,000×1.012×1.012×1.012×...×1.012 x times after x years.
and we get
y = 345,000 × (1.012)^x
The red rectangle is the pre-image and the green rectangle is the image. What would be the coordinate of A" if the scale factor of 3 is used?
Pls show all your work!
Keep in mind I will immediately mark brainliest for the right answer!
The scale factor that has been used is 1/2
How to determine what scale factor has been used?From the question, we have the following parameters that can be used in our computation:
Side length on red = 6 units
Corresponding side length on green = 3 units
Given that the red rectangle is the pre-image and the green rectangle is the image
We have
Scale factor = Green/Red
So, we have
Scale factor = 3/6
Evaluate
Scale factor = 1/2
Hence, the scale factor is 1/2
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Identify the Property Below:
ab² • 0 = 0 ______
(7 + 5) + 1 = 7 + (5 + 1) _____
6 = 6 ____
7(x-3) = 7x - 21 ______
x + (-x) = 0 ___
if y = 3, then 3 = y ___
if x = -1, and -1 = z, then x = z ____
4x • 1 = 4x ____
(a+b) + 0 = (a+b) ____
3 • 1/3 = 1 ___
The properties are listed below.
What is properties of multiplication and addition?Properties of addition and multiplication are defined for the various conditions and rules of addition and multiplication. The properties are:
Commutative propertyAssociative Property Distributive Property Identity Propertyab² • 0 = 0The product between any number and this one is zero. This comes from the existence of zero, which states that there must exist a value that represents nothingness, the zero.
(7 + 5) + 1 = 7 + (5 + 1)This is the associative property of addition,
6 = 6This is the reflexive property of equality, which says that every number is equal to itself.
7(x-3) = 7x - 21This is the distributive property of addition which says ,
C*(A + B) = C*A + C*B
x + (-x) = 0This is the inverse property of addition, this property says that for any real number x there exists a real number -x such that x+(-x)=0.
if y = 3, then 3 = yThis is the symmetric property.
if x = -1, and -1 = z, then x = zThis is the transitive property, we can apply it in the next way
then x = z
4x • 1 = 4xThis is the identity property of multiplication.
(a+b) + 0 = (a+b)This is identity property of addition is that when a number n is added to zero, the result is the number itself i.e. n + 0 = n.
3 • 1/3 = 1This is inverse property of multiplication. It states that if you multiply a number by its reciprocal, also called the multiplicative inverse, the product will be 1
Hence, the properties are identified.
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Jana is wondering about the probabilities of genetic crosses. She knows that
when her gerbil had babies, there was a 50% chance of babies being male
and a 50% chance of being female. Her gerbil gave birth to 6 babies, and 4
were girls and 2 were boys. Why weren't they 50% male and 50% female?
Experimental Probability
Just because the odds are 50% for both genders doesn't mean there is any reason for it to be even. It's like flipping a coin six times. there's no reason for it to land on tails 3 times and heads 3 times. Its just more likely
Please solve quickly! Within 30 minutes would be great!
Please solve for the variable indicated.
A=1/2h(b+B), solve for b
If you could break it down step by step that would be super helpful! I’m very confused. Thank you!
Answer:
Solve for B:
[tex]B = -b+\frac{5A}{6h}[/tex]
Step-by-step explanation:
[tex]A=1.2h(b+B)[/tex]
Use the distributive property to multiply [tex]1.2h by b+B[/tex]
[tex]A=1.2hb+1.2hB[/tex]
Swap sides so that all variable terms are on the left hand side.
[tex]1.2hb+1.2hB=A[/tex]
Subtract [tex]1.2hb[/tex] from both sides.
[tex]1.2hB= A - 1.2hb[/tex]
The equation is in standard form.
[tex]\frac{6h}{5} B=-\frac{6bh}{5} +A[/tex]
Divide both sides by [tex]1.2h[/tex].
[tex]\frac{5*(\frac{6h}{5})B }{6h} =\frac{5(-\frac{6bh}{5}+A) }{6h}[/tex]
Dividing by [tex]1.2h[/tex] undoes the multiplication by [tex]1.2h[/tex].
[tex]B = \frac{5(-\frac{6bh}{5}+A) }{6h}[/tex]
Divide [tex]A - \frac{6hb}{5}[/tex] by [tex]1.2h[/tex].
[tex]B = -b+\frac{5A}{6h}[/tex]
Solve For B:
[tex]B = -b+\frac{5A}{6h}[/tex]
Apply the square root principle to solve (x – 3)2 + 9 = 0.
a. x = 3 + 3i, 3 – 3i
b. x = 0, –6
c. x = 0, 6
d. x = –3 + 3i, –3 – 3i
Answer:
Step-by-step explanation:
First, let's simplify the equation:
(x – 3)² + 9 = 0
(x – 3)² = -9
Now we can apply the square root principle:
x – 3 = ±√(-9)
x – 3 = ±3i
Solving for x:
x = 3 ± 3i
Therefore, the answer is (a) x = 3 + 3i, 3 – 3i.
MATH QUESTION <3 PLEASE HELP
The angle that the flagpole makes with the ground is 20°.
What are trigonometric identities?There are three commonly used trigonometric identities.
Sin x = 1/ cosec x
Cos x = 1/ sec x
Tan x = 1/ cot x or sin x / cos x
Cot x = cos x / sin x
We have,
Flagpole height = 16 feet
From figure B,
The angle of elevation has increased this means,
The flagpole is leaning.
Now,
The angle that the flagpole makes with the ground.
Cos m = 15/16
Cos m = 0.9375
m = [tex]cos^{-1}[/tex] 0.9375
m = 20.36
m = 20°
Thus,
The angle that the flagpole makes with the ground is 20°.
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-3 (3b + 6) = 36 - 3b
Answer:
b = -9
Step-by-step explanation:
-3 (3b + 6) = 36 - 3b
-9b - 18 = 36 - 3b
-6b - 18 = 36
-6b = 54
b = -9
Let's check
-3(3(-9) + 6) = 36 - 3(-9)
-3( -27 + 6) = 36 + 27
-3( -21) = 63
63 = 63
So, b = -9 is the correct answer.
Step-by-step explanation:
-3(3b+6)=36-3b
-9b-18=36-3b
-9b+3b=36+18
-6b=54
divide both sides by -6
b= -9
A bacteria population starts at 2,032 and decreases at about 15% per day. Write a function representing the number of bacteria present each day. Graph the function. After how many days will there be fewer than 321 bacteria?
The function representing the number of bacteria present each day is f(t) = 2032[tex](0.85)^{t}[/tex].
What is meant by function?Numbers, formulae, and related structures, shapes, and the spaces they occupy are all issues in the field of mathematics, as are quantities and their variations. f(x) = x2 is a prime example of a straightforward function. The function in this equation is called f(x), and it squares the value of "x". Assume that f(3) = 9 if, for example, x = 3. Several other functions include f(x) = sin x, f(x) = x2 + 3, f(x) = 1/x, f(x) = 2x + 3, etc.Each element of X receives the exact same number of elements from Y when a function from one set to the other is used.Both the set X and the set Y are referred to as the function's domain and codomain, respectively.
Beginning in 2032, there will be a 15% daily decline in the number of microorganisms.
a function that displays the daily average amount of microorganisms.
f(x) = 2032[tex](1 - .15)^{t}[/tex]
f(x) = 2032[tex](0.85)^{t}[/tex]
Therefore f(t) = 2032[tex]0.85^{t}[/tex] is the correct answer.
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a learning experiment requires a rat to run a maze (a network of pathways) until it locates one of three possible exits. exit 1 presents a reward of food, but exits 2 and 3 do not. (if the rat eventually selects exit 1 almost every time, learning may have taken place.) let yi denote the number of times exit i is chosen in successive runnings. for the following, assume that the rat chooses an exit at random on each run. (a) find the probability that n
The probability that when n=6, [tex]Y_{1}[/tex] =3, [tex]Y_{2}[/tex]=1, [tex]Y_{2}[/tex]=2 is 0.0822.
We have to find the probability that n=6, [tex]Y_{1}[/tex] =3, [tex]Y_{2}[/tex]=1, [tex]Y_{2}[/tex]=2
A rat has three possible exits to come out from the maze
The pmf of multinomial distribution is
P[tex](y_{1}, y_{2}, y_{3}......, y_{k} ) = \frac{n!}{{y_{1!} }y_{2}!....y_{k}! } p^y_1 1p^y_2 2......p^y_k k[/tex]
The probability of choosing one of the ways is 1/3
Then [tex]p_{1}[/tex]= 1 / 3,[tex]q_{1}[/tex] = 1-(1/3) = 2/3
[tex]p_{2} = 1/3, q_{2} = 2/3\\ p_{3} = 1/3, q_{3} = 2/3[/tex]
[tex]P(y_{1}, y_{2}, y_{3} ) = P(1,2,3)\\ = \frac{6!}{3! 2! 1!} (\frac{1}{3} )3(\frac{1}{3} )2(\frac{1}{3} )1[/tex]
=60(0.00137)
=0.0822
Hence, the probability that when n=6, [tex]Y_{1}[/tex] =3, [tex]Y_{2}[/tex]=1, [tex]Y_{2}[/tex]=2 is 0.0822.
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The correct question is:
A learning experiment requires a rat to run a maze (a network of pathways) until it locates one of three possible exits. Exit 1 presents a reward of food, but exits 2 and 3 do not. (If the rat eventually selects exit 1 almost every time, learning may have taken place.) Let [tex]Y_{i}[/tex] denote the number of times exit i is chosen in successive runnings. For the following, assume that the rat chooses an exit at random on each run.
a Find the probability that n = 6 runs result in [tex]Y_{1} = 3, Y_{2} = 1, and Y_{3} = 2.[/tex]
1. Find the 15th term in the sequence if a1= 3 and d= 4
2. Find Sn for the arithmetic series where a1= 5, an= 119, n= 20
3. Find Sn for the arithmetic series where a1= 12, d= 6, n= 15
4. Find the 6th term in the geometric sequence where a1= 2, a6= 64, r= 2
5. Find Sn for the geometric series where a1= 2 , r= 4, n= 6
an = 3 + (15 - 1) * 4 = 3 + 14 * 4
= 3 + 56
= 59
So, the 15th term in the sequence is 59.
2. To find the sum of an arithmetic series with first term a1, last term an, and number of terms n, we use the formula: Sn = n/2 * (a1 + an). Plugging in the values, we get:
Sn = 20/2 * (5 + 119)
= 20/2 * 124
= 620
So, the sum of the series is 620.
3. To find the sum of an arithmetic series with first term a1, common difference d, and number of terms n, we use the formula: Sn = n/2 * (2a1 + (n - 1)d). Plugging in the values, we get:
Sn = 15/2 * (2 * 12 + (15 - 1) * 6) = 15/2 * (24 + 84)
= 15/2 * 108
= 810
So, the sum of the series is 810.
4.To find the nth term in a geometric sequence with first term a1, common ratio r, and nth term an, we use the formula: an = a1 * r⁽ⁿ⁻¹⁾. Plugging in the values, we get:
64 = 2 * r⁽⁶⁻¹⁾
64 = 2 * r⁵
32 = r⁵
r = [tex]2^{(5^{(1/5)} )}[/tex]
So, the 6th term in the sequence is 64.
5. To find the sum of a finite geometric series with first term a1, common ratio r, and number of terms n, we use the formula: Sn = a1 * (1 - rⁿ) / (1 - r). Plugging in the values, we get:
Sn = 2 * (1 - 4⁶) / (1 - 4)
= 2 * (1 - 4096) / -3
= 2 * (-4095) / -3
= 2730
So, the sum of the series is 2730.
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the box-and-whisker plot shows the number of times students bought lunch in a given month at the school cafeteria. a box and whisker plot with minimum 4, first quartile 11, median 14, third quartile 16, and maximum 20 what is the interquartile range of the data? provide your answer below:
The interquartile range of the data is
The interquartile range (IQR) is a measure of variability in a dataset that is calculated as the difference between the third quartile (Q3) and the first quartile (Q1).
The IQR is the range of the middle 50% of the data, and it provides a measure of the spread of the central part of the distribution.
From the given information, we have:
Minimum value = 4
Q1 = 11
Median = 14
Q3 = 16
Maximum value = 20
To find the IQR, we subtract Q1 from Q3:
IQR = Q3 - Q1 = 16 - 11 = 5
Therefore, the interquartile range of the data is 5.
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Help me with this worksheet please
The trigonometry ratio values are tan(50) = 1.19 and cos(70) = 0.34
The trigonometry ratio valuesThis can be calculated using a calculator
So, we have
tan(50) = 1.19 and cos(70) = 0.34
The measures of the anglesThis can be calculated using a calculator
So, we have
sin(84) = 0.9945 and tan(75) = 3.7321
The trigonometry ratio valuesHere, we make use of the laws of cosines and sines
So, we have
cos(X) = 15/17 = 0.8824
sin(C) = 36/45 = 0.8
The measures of the indicated anglesHere, we make use of the laws of tangents and sines
So, we have
tan(?) = 6/13 = 0.4615
? = 24.77
sin(?) = 41/59 = 0.6949
? = 44.01
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Let V be the space of all infinite sequences of real num- bers. See Example 5. Which of the subsets of V given in Exercises 12 through 15 are subspaces of V? 12. The arithmetic sequences [i.e. sequences of the form (a,a + k;a + 2k,u 3k for some constants CL and k]
Yes , the arithmetic sequence ( written as sequences of the form (a , a+k , a+2k , a+3k for some constants a and k ) is a subspace of V .
To determine whether subsets of V in the arithmetic sequences are subspaces of V, we check if they satisfy the three conditions for a subset to be a subspace:
(i) The subset must contain the zero vector.
(ii) The subset must be closed under vector addition.
(iii) The subset must be closed under scalar multiplication.
Let S be a subset of V consisting of arithmetic sequences.
Now we check the three conditions :
(i) We find an arithmetic sequence in S that has all its terms equal to zero. The only arithmetic sequence that satisfies this is the sequence (0, 0, 0, 0, ...), which is in S.
So , the subset "S" contains zero vector.
(ii) Next we need to show that if u and v are two arithmetic sequences in S, then their sum "u + v" is also an arithmetic sequence in S.
Let U = (a, a+k, a+2k, a+3k, ...) and V = (b, b+l, b+2l, b+3l, ...) be two arithmetic sequences in S.
Then, their sum is ⇒ u+v = (a+b, a+k+b+l, a+2k+b+2l, a+3k+b+3l, ...).
This is also an arithmetic sequence,
So , subset "S" is closed under vector addition.
(iii) Next , we show that if U is an arithmetic sequence in S and C is a scalar, then CU is also an arithmetic sequence in S.
Let U = (a, a+k, a+2k, a+3k, ...) be an arithmetic sequence in S, and let c be a scalar.
we get , CU = (ca, ca+ck, ca+2ck, ca+3ck, ...) is also an arithmetic sequence,
So , subset "S" is closed under scalar multiplication.
we see that , S satisfies all three conditions, it is a subspace of V.
Therefore, any subset of V consisting of arithmetic sequences is a subspace of V.
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The given question is incomplete , the complete question is
Let V be the space of all infinite sequences of real numbers , Which of the subsets of V in The arithmetic sequences [i.e. sequences of the form (a , a+k , a+2k , a+3k for some constants a and k] are subspaces of V ?
A machine must produce a bearing that is within 0.02 inches of the correct diameter 4.1 inches. using x as the diameter of the bearing, write this statement using absolute value notation.____ ___ Preview
A machine must produce a bearing that is within 0.02 inches of the correct diameter 4.1 inches. By writing this statement using absolute value notation is:
| x - 4.1 | ≤ 0.02
This notation expresses the requirement that the diameter of the bearing, represented by x, must be within 0.02 inches of the correct diameter of 4.1 inches. The absolute value of the difference between x and 4.1 must be less than or equal to 0.02 for the bearing to meet the specification.
The absolute value function is used to ensure that the difference between x and 4.1 is always positive, regardless of whether x is greater than or less than 4.1. This is important because the requirement is concerned with the magnitude of the difference, rather than its sign. If the absolute value of the difference is less than or equal to 0.02, then the diameter of the bearing is considered to be within the required tolerance.
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