The decay function d(x) is an exponential function with an initial value of 850, a decay factor of 0.94, and a rate of decay that increases as x increases. The function has a horizontal asymptote at y=0, and its domain is all real numbers while its range is (0, 850].
The decay function d(x) can be described by the following characteristics:Exponential Decay: The function d(x) is an exponential function because it has a constant base (0.94) raised to a variable exponent (x).
Initial Value: The initial value of the function d(x) is 850, which represents the value of the function when x=0.
Decay Factor: The decay factor of the function d(x) is 0.94, which is less than 1. This means that as x increases, the function decreases and approaches zero, but never reaches zero.
Rate of Decay: The rate of decay of the function d(x) is determined by the value of the decay factor, which is 0.94. The closer the decay factor is to 1, the slower the rate of decay. Conversely, the closer the decay factor is to 0, the faster the rate of decay.
Asymptote: The function d(x) has a horizontal asymptote at y=0. This means that as x becomes very large, the function approaches but never touches the x-axis.
Domain and Range: The domain of the function d(x) is all real numbers, and the range is (0, 850]. This means that the function outputs a positive value less than or equal to 850, but never outputs zero.
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PLEASE HELP!!! WILL MARK BRANLIEST!!!
Answer:
The point z = 3+4i is plotted as a blue dot, and the two square roots are plotted as a red dot and a green dot. The magnitudes of z and its square roots are shown by the radii of the circles centered at the origin.
Step-by-step explanation:
qrt(z) = +/- sqrt(r) * [cos(theta/2) + i sin(theta/2)]
where r = |z| = magnitude of z and theta = arg(z) = argument of z.
Calculate the magnitude of z:
|r| = sqrt((3)^2 + (4)^2) = 5
And the argument of z:
theta = arctan(4/3) = 0.93 radians
Now, find the two square roots of z:
sqrt(z) = +/- sqrt(5) * [cos(0.93/2) + i sin(0.93/2)]
= +/- 1.58 * [cos(0.47) + i sin(0.47)]
= +/- 1.58 * [0.89 + i*0.46]
Using a calculator, simplify this expression to:
sqrt(z) = +/- 1.41 + i1.41 or +/- 0.2 + i2.8
find the volume v of the solid obtained by rotating the region bounded by the given curves about the specified line. calculatot
Therefore, the volume of the solid obtained by rotating the region bounded by the curves y = x^2, y = 0, x = 0, and x = 1 about the line x = 2 is (5/6)π cubic units.
To find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line, we use the method of cylindrical shells. The formula for the volume of a cylindrical shell is given as,
Volume = 2πrh * Δx
Where, r is the distance of the shell from the axis of rotation, h is the height of the shell, and Δx is the thickness of the shell. The factor 2π accounts for the entire circumference of the shell.
Example problem: Find the volume of the solid generated by rotating the region bounded by the curves y = x^2, y = 0, x = 0, and x = 1 about the line x = 2.
Solution:
Step 1: Draw the graph of the region to be rotated
Step 2: Identify the shell radius and height
For a shell at position x, the radius is given by r = 2 - x (the distance of the line x = 2 from the axis of rotation)
The height of the shell is given by h = x^2 (the difference between the top and bottom curves)
Step 3: Write the volume formula
Volume = 2πrh * Δx
Step 4: Integrate to find the total volume
The limits of integration are from 0 to 1 since the curves intersect at (0,0) and (1,1).
[tex]∫ 2π(2 - x)(x^2) dx from 0 to 1[/tex]
= [tex]2π [∫ 2x^2 - x^3 dx from 0 to 1][/tex]
= [tex]2π [(2/3) - (1/4)][/tex]
= 2π [5/12][tex]2π [5/12][/tex]
= (5/6)π cubic units
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A rectangle has length 64mm and 37mm each correct to the nearest millimetre.
(a) Write down the lower bound for the length.
(b) Calculate the lower bound for the perimeter of the rectangle.
Step-by-step explanation:
(a) To find the lower bound for the length of the rectangle, we need to subtract half of the smallest unit of measurement from the given measurement.
The smallest unit of measurement given is 1 mm, so half of that is 0.5 mm.
Therefore, the lower bound for the length of the rectangle is:
64 mm - 0.5 mm = 63.5 mm
(b) The perimeter of a rectangle is calculated by adding up the length of all four sides.
The lower bounds for the length and width of the rectangle are 63.5 mm and 36.5 mm, respectively.
So, the lower bound for the perimeter is:
2(63.5 mm + 36.5 mm) = 2(100 mm) = 200 mm
Answer and explanation below. Credits to Chris, Mechanical engineering college, Master's degree
Consider the following exponential probability density function. f(x) = 1/3 4 e^-x/3 for x > 0 a. Write the formula for P(x < x_0). b. Find P(x < 2). c. Find P(x > 3). d. Find P(x < 5). e. Find P(2 <.x <5).
The probability that x is less than 2 is approximately 0.4866. The probability that x is greater than 3 is approximately 0.3528. The probability that x is less than 5 is approximately 0.6321. The probability that x is between 2 and 5 is approximately 0.1455.
The given probability density function is an exponential distribution with a rate parameter of λ = 1/3. The formula for P(x < x_0) is the cumulative distribution function (CDF) of the exponential distribution, which is given by:
F(x_0) = ∫[0,x_0] f(x) dx = ∫[0,x_0] 1/3 * 4 * e^(-x/3) dx
a. Write the formula for P(x < x_0):
Using integration, we can solve this formula as follows:
F(x_0) = [-4e^(-x/3)] / 3 |[0,x_0]
= [-4e^(-x_0/3) + 4]/3
b. Find P(x < 2):
To find P(x < 2), we simply substitute x_0 = 2 in the above formula:
F(2) = [-4e^(-2/3) + 4]/3
≈ 0.4866
Therefore, the probability that x is less than 2 is approximately 0.4866.
c. Find P(x > 3):
To find P(x > 3), we can use the complement rule and subtract P(x < 3) from 1:
P(x > 3) = 1 - P(x < 3) = 1 - F(3)
= 1 - [-4e^(-1) + 4]/3
≈ 0.3528
Therefore, the probability that x is greater than 3 is approximately 0.3528.
d. Find P(x < 5):
To find P(x < 5), we simply substitute x_0 = 5 in the above formula:
F(5) = [-4e^(-5/3) + 4]/3
≈ 0.6321
Therefore, the probability that x is less than 5 is approximately 0.6321.
e. Find P(2 < x < 5):
To find P(2 < x < 5), we can use the CDF formula to find P(x < 5) and P(x < 2), and then subtract the latter from the former:
P(2 < x < 5) = P(x < 5) - P(x < 2)
= F(5) - F(2)
= [-4e^(-5/3) + 4]/3 - [-4e^(-2/3) + 4]/3
≈ 0.1455
Therefore, the probability that x is between 2 and 5 is approximately 0.1455.
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An experiment was conducted to test the effect of a new cholesterol medicine. Fifteen people were treated with medicine X and 15 with medicine y for a month; then the subjects' cholesterol level was measured. A
significance test was conducted at the 0. 01 level for the mean difference in cholesterol levels between medicine X and medicine Y. The test resulted in t - 2. 73 and p = 0. 3. If the alternative hypothesis in
question was Mo: px - WY* 0, where wx equals the mean cholesterol level for subjects taking medicine X and by equals the mean cholesterol level for subjects taking medicine Y, what conclusion can be drawn? (2
points)
There is sufficient evidence that there is a difference in mean cholesterol level when using medicine X and medicine Y.
There is not a significant difference in mean cholesterol level when using medicine X and medicine Y.
There is sufficient evidence that medicine X towers cholesterol levels more than medicine Y.
The proportion of subjects who lowered their cholesterol level with medicine X is greater than the proportion of subjects who lowered their cholesterol level with medicine Y.
There is insufficient evidence that the proportion of people who lowered cholesterol level on medicine X and Y is different.
The right answer is provided by: Taking into account the hypothesis tested and the p-value of the test.
There is enough proof to conclude that using drugs X and Y results in different mean cholesterol levels.
What theories are being tested?
If there is no difference in the cholesterol levels, then the null hypothesis is tested, which is:
H₀:Hₓ-H
It is determined whether there is a difference at the alternative hypothesis, so:
H₀:Hₓ-H≠0
The p-value of 0.003 0.01 leads us to reject the null hypothesis and determine that there was a difference, hence the following is the right response:
There is enough proof to conclude that using drugs X and Y results in different mean cholesterol levels.
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Solve the equation. |k+ 7| = 3
(-4, 10)
{all real numbers greater than or equal to -10 and less than or equal to -4}
{-4,4)
{-10, -4}
Answer:
-10 or -4
Step-by-step explanation:
the absolute value of -10 + 7 is 3
the absolute value of -4 + 7 is 3
absolute value just removes the negative sign of what is inside of it.
Which of the following algorithms sorts data by recursively splitting the array into smaller sub-arrays, until eventually reaching single-element sections, and then joins the sub-arrays back together in sorted order? A Heap Sort B Quick Sort Merge Sort (D Radix Sort
B) Quick Sort is algorithms sorts data by recursively splitting the array into smaller sub-arrays, until eventually reaching single-element sections, and then joins the sub-arrays back together in sorted order
What is Merge Sort Algorithm?Merge Sort is an effective algorithm for sorting the data items of a list or an array. This algorithm's fundamental idea is to break the unsorted list into many sub-lists until every sub-list contains only one item.
The merging procedure then recombines all of the sublists, resulting in a sorted list. Divide-and-conquer is the name given to this merging process. Merge Sort divides the original list into N sublists, each of which contains one element.
The Merge Sort algorithm is faster than other sorting algorithms when it comes to time complexity. The time complexity of Merge Sort is O (n log n), which is better than Bubble Sort and Insertion Sort.
It is a more efficient sorting algorithm than other algorithms, such as bubble sort, which has a higher time complexity of O (n²).
Therefore, we can say that Merge Sort is the most efficient algorithm to sort an array in ascending or descending order.
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The cost price of a book is Rs 20 . It is sold at 10% profit. Find its sale price
Answer:
Cost price (CP) = Rs20
% Profit = 10%
Selling price (SP) = y
By formula:
% Profit = (Profit/CP) x 100%
But Profit= SP - CP
Therefore;
% Profit = [(SP-CP)/C.P)] x 100%
Substituting, we have:
10% = [(y-20)/20] x 100%
10/100 = (y-20)/20
1/10 = (y-20)/20
1 = (y-20)/2
Cross multiply
y - 20 = 2
y = 2 + 20
y = 22
Therefore, the Sale price is Rs 22
Which of the following examples satisfy the hypotheses of the Extreme Value Theorem on the given interval?
A. f(x)=1/x on −10≤x≤10
B. g(x)=6x^2+3 on 0≤x≤4
C. k(x)={3x^2+9 for 0≤x<2, 12x for 2≤x≤10} on 0≤x≤10
D. h(x)=(e^x)/x on 2≤x≤16
E. m(x)=6x^3+x+1 on −4
The function that satisfies the hypotheses of the Extreme Value Theorem on the given interval is given by
B. g(x)=6x^2+3 on 0≤x≤4
D. f(x) = (e^x)/x for 2 ≤ x ≤ 16.
Step 1: State the Extreme Value Theorem
The Extreme Value Theorem states that if a function is continuous on a closed interval [a,b], then the function must have a maximum and a minimum on the interval.
Step 2: Check for continuity and closed interval for each function
A. f(x) = 1/x on −10 ≤ x ≤ 10
The function f(x) = 1/x is continuous on the interval (-10, 0) and (0, 10).
However, since the interval given is [−10, 10], we see that the function is not continuous over the closed interval.
Hence the function does not satisfy the hypotheses of the Extreme Value Theorem.
B. g(x) = 6x^2+3 on 0 ≤ x ≤ 4
The function g(x) is continuous on the interval [0, 4].
Therefore, this function satisfies the hypotheses of the Extreme Value Theorem on the given interval.
C. k(x) = {3x^2+9 for 0 ≤ x < 2, 12x for 2 ≤ x ≤ 10} on 0 ≤ x ≤ 10
The function k(x) is continuous on the interval [0, 2) and (2, 10]. H
However, since the interval given is [0, 10], we see that the function is not continuous over the closed interval.
Hence the function does not satisfy the hypotheses of the Extreme Value Theorem.
D. h(x) = (e^x)/x for 2 ≤ x ≤ 16The function h(x) is continuous on the interval [2, 16].
Therefore, this function satisfies the hypotheses of the Extreme Value Theorem on the given interval.
E. m(x) = 6x^3+x+1 on −4 < x < 3
The function m(x) is continuous on the interval (-4, 3).
However, since the interval given is [-4, ∞), we see that the function is not continuous over the closed interval.
Hence the function does not satisfy the hypotheses of the Extreme Value Theorem.
Therefore, the only function that satisfies the hypotheses of the Extreme Value Theorem on the given interval is given by
B. g(x)=6x^2+3 on 0≤x≤4
D. f(x) = (e^x)/x for 2 ≤ x ≤ 16
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Move the manilla point as close to the circle as possible so that the blue arc almost disappears keep the manilla point on the circle
What previously learned theorem do these transformations reveal
A theorem which these transformations reveal include the following: theorem of intersecting secants.
What is the theorem of intersecting secants?In Mathematics and Geometry, the theorem of intersecting secants states that when two lines intersect outside a given circle, the measure of the angle formed by these intersecting lines is equal to one-half (½) of the difference of the two (2) arcs it intercepts.
By applying the theorem of intersecting secants, the value of any of the angle subtended by the intersecting lines can be calculated by using the following mathematical equation:
m∠a = One-half(y – x).
m∠a = ½(y – x).
Where:
x and y represent the angles formed by the intersecting lines.
Therefore, the theorem of intersecting secants describe these set of transformations.
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PLEASE HELP a) Calculate the scale factor from shape A to shape B.
b) Find the value of t.
Give each answer as an integer or as a fraction (preferable) in its simplest form.
Question attached, Thank you!
Answer:
see explanation
Step-by-step explanation:
(a)
the scale factor is the ratio of corresponding sides, shape B to shape A
scale factor = [tex]\frac{12}{15}[/tex] = [tex]\frac{4}{5}[/tex]
(b)
t is then [tex]\frac{4}{5}[/tex] of 7 = [tex]\frac{4}{5}[/tex] × 7 = [tex]\frac{28}{5}[/tex]
The table below shows the heights, in metres
(m), of three different plants.
Put the plants into ascending order based on
their height.
Plant:
Rose=59/100
Barberry=7/10
Holly:11/20
Answer please
Answer:
7/10
59/100
11/20
Step-by-step explanation:
59/100=59/100
7/10=70/100
11/20=55/100
Y=5x+17 Y=-2x+4 solve with substitution
Answer:
x = -13/7, y = 54/7
Step-by-step explanation:
- Set equations equal to each other
[tex]5x + 17 = -2x +4[/tex]
Add 2x to both sides
[tex]7x +17 = 4[/tex]
Subtract 17 from both sides
[tex]7x = -13[/tex]
divide both sides by 7
[tex]x = -13/7[/tex]
Then substitute x into one of the equations. I picked the first one.
[tex]y = 5 (-13/7) + 17[/tex]
I multiplied 5 by -13 to get -65/7. Then I turned 17 into sevenths: 17 x 7 = 119, so that's 119/7. 119 + -65 = 54, so 54/7 for y.
Let me know if this helped by hitting thanks or brainliest. If not, please comment and I'll get back to you ASAP!
Can someone please help me with this?
The measure of the side MR is given as 20
How to solve for the side MRWe have to first find the value of the angle
∠TQR = 180 - (35 + 25)
= 180 - 60
= 120 degrees
180 - 120 = 60 degrees
angle TPR = 90 degrees
next we have to find ∠PTQ
= 180 - (90 + 60)
= 180 - 150
= 30 degrees
given that TMN = TQR
TNP = PTQ
So if TMN = 35 degrees since TQR = 35 degrees
PTQ = 30, so TNP = 30 degrees
The measure of QR = 4 since MN = 4
NP = pq
np = 6
Hence the measure of MR
= 4 + 6 + 6 + 4
= 20
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a family of five recently replaced its 5-gallon-per-minute showerheads with water-saving 2-gallon-per-minute showerheads. each member of the family averages 8 minutes in the shower per day. in a 30-day period, how many fewer gallons of water will the family use with the new showerheads? responses 60 60 800 800 2,400 2,400 3,600 3,600 7,200
The family will use 3,600 fewer gallons of water with the new showerheads in a 30-day period.
A family of five recently replaced its 5-gallon-per-minute showerheads with water-saving 2-gallon-per-minute showerheads. Each member of the family averages 8 minutes in the shower per day. In a 30-day period, the family will use 3,600 fewer gallons of water with the new showerheads. How many fewer gallons of water will the family use with the new showerheads?
Calculate the number of gallons used per person per shower before:
5 gal/min x 8 min = 40 gallons per person per shower.
The number of gallons used per person per shower after:
2 gal/min x 8 min = 16 gallons per person per shower.
The number of gallons used per person per day before:
40 gallons x 1 shower = 40 gallons per person per day.
The number of gallons used per person per day after:
16 gallons x 1 shower = 16 gallons per person per day
The number of gallons used per family per day before:
40 gallons x 5 people = 200 gallons per day
The number of gallons used per family per day after:
16 gallons x 5 people = 80 gallons per day
The number of gallons used per month before:
40 gallons x 5 people x 30 days = 6,000 gallons per month
The number of gallons used per month after:
16 gallons x 5 people x 30 days = 2,400 gallons per month
The family will use 6,000 - 2,400 = 3,600 fewer gallons of water with the new showerheads in a 30-day period. Therefore, the answer is 3,600.
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Find the value of X.
Numbers1,3,4,5,8,9
The value of x for each circle is calculated as follows:
1. x = 148°
3. x = 82°
4. x = 135°
5. x = 7
8. x = 6
9. x = 13
Define the term Circle identities?The Circle identities refer to a set of fundamental identities in trigonometry that relate the six trigonometric functions of an angle in a right-angled triangle.
1. Given, SR = ST
then, arcSR = arcST = x°
and arcRT = 64°
Sum of arc,
x + x +64 =360
x = 148°
3. Given, JK=LM
then, arcJK=arcLM
So, x=82°
4. Given, BA=AC =9
Then, arcBA = arcAC = x°
and arcBC = 90°
Sum of arc,
x + x +90 =360
x = 135°
5. Given, arc are equal then side also equal
2x+4=18
x=7
8. Given that, Circle M similar to circle P, and arc are equal so, side also equal
2x+24=6x
x=6
9. Given that, Circle V similar to circle W,
arcYZ + 198 =360
arcYZ = 162°
Now, arcTU=arcYZ
Then, 9x-78 = 3x
So, x=13
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Need help with math
The new coordinates of the two points after rotating the parallel lines 180 degrees clockwise are (2, -7) and (-8, 5).
What are parallel lines?
In geometry, parallel lines can be defined as two lines in the same plane that are at equal distance from each other and never meet.
If the set of parallel lines contains the points (-2, 7) and (8, -5), then the two lines are parallel to each other and have the same slope. We can find the slope of the line that passes through these two points using the slope formula:
slope = (y2 - y1) / (x2 - x1)
slope = (-5 - 7) / (8 - (-2))
slope = -12 / 10
slope = -6 / 5
So, the equations of the two parallel lines are:
y - 7 = (-6 / 5)(x + 2) --- equation 1
y + 5 = (-6 / 5)(x - 8) --- equation 2
To rotate the lines 180 degrees clockwise, we need to negate both the x and y coordinates of the points on the lines. That is, we need to replace each point (x, y) with the point (-x, -y).
So, after the rotation, the new coordinates of the two points will be:
(-(-2), -7) = (2, -7) --- for point (-2, 7)
(-(8), -(-5)) = (-8, 5) --- for point (8, -5)
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Question 1 0.5 pts A man walks along a straight path at a speed of 4 ft/s. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. Place the steps below in the correct order that they should be performed in order to the determine the rate at which the searchlight is rotating when the man is 15 ft from the point on the path closest to the searchlight. 1. Step 1 Draw a picture 2. Step 2 Write down the numerical info 3. Step 3 Determine what you are asket 4. Step 4 Write an equation relating the 5. Step 5 Plug in your known informatic 6. Step 6 Differentiate both sides of the h at a speed of 4 ft/s. A searchlight is located on the ground 20 ft d on th [Choose ] ey the de Differentiate both sides of the equation with respect to t on the 1 Determine what you are asked to find Write down the numerical information that you know Write an equation relating the variables Draw a picture Plug in your known information to solve the problem Write down the numerical info v V Determine what you are asker
In order to determine the rate at which the searchlight is rotating when the man is 15 ft from the point on the path closest to the searchlight, the following steps should be performed in this order
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0 | 40
1 | 40.75
3 | 42.5
5 | 44
7 | 45.25
Use the line segment that connects ______ and _______ to estimate the vine length after 4 days.
After 4 days, the vine length is about ______.
The top is a chart Please help me!!!
Use the line segment that connects (3, 42.5) and (5, 44) to estimate the vine length after 4 days.
How to determine the lengths after 4 daysGiven the table of values
To do this, we make use of linear interpolation
Such that the points closest to day 4 are (3, 42.5) and (5, 44)
The slope of the line passing through the two points can be found using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (3, 42.5) and (x2, y2) = (5, 44).
Thus, we have:
m = (44 - 42.5) / (5 - 3) = 0.75
So, we have
y = 0.75x + b
Using the point (3, 42.5), we get:
42.5 = 0.75(3) + b
Solving for b, we get:
b = 42.5 - 0.75(3)
b = 40.25
So, the equation is
y = 0.75x + 40.25
To estimate the vine length after 4 days, we substitute x = 4 into the equation and solve for y:
y = 0.75 * 4 + 40.25
y = 43.25
Therefore, the vine length after 4 days is estimated to be about 43.25 units.
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a) What is the mean proportional between 3 and 12?
b) What is the geometric mean of 3 and 12?
Answer:
a. 6
b. 6
Step-by-step explanation:
A. mean proportion
3:x::x:12
product of mean = product of extreme
3 x 12 = x^2
x^2= 36
now take square root on both sides
x = 6
as sqaure root will cancel the square on x and square root of 36 is 6.
B. the geometric mean is the product of all
the numbers in a set, with the root of how many numbers there are. here, there are two numbers, so a square root is used. the value of the geometric mean of 3 and 12 is 6.
Hope's it helps...
24 354÷5 (use long division)
Please help me guys i need this right now
Answer:
487
or
4870
Step-by-step explanation:
(I might not have done it right)
is it 24,354 / 5 or 2,435 / 5?
Because I got 487
But to check an answer you must multiply
When I multiplied 5 x 487 the answer is 2,435
But 5 x 4870 is 24,350
It has to be one of these
just need help with part (b) please
In response to the stated question, we may state that As a result, the formula for the sequence's nth term is: [tex]a_n = a _1 * r^{(n-1) (n-1)}[/tex]
What is Sequences?In mathematics, a sequence is an ordered list of items. Elements can be numbers, functions, or other mathematical objects. A series is commonly expressed by putting the phrases in parentheses and separating them with commas. A natural number series, for example, can be denoted as: (1, 2, 3, 4, 5, ...) Similarly, the even number series is labelled as follows: (2, 4, 6, 8, 10, ...) A series can be finite or infinite depending on whether it has a finite or infinite number of words.
a. Since the difference between subsequent terms is constant, the series 5, 7, 8,... is an arithmetic sequence. The usual difference is 2, specifically. As a result, the formula for the sequence's nth term is:
[tex]a_n = a_1 + (n-1)d[/tex]
where a 1 represents the first term, d represents the common difference, and n represents the term number. Using the provided values, we get:
[tex]a_n = 5 + (n-1)2\\a_n = 2n + 3[/tex]
b. Since the ratio between subsequent terms is constant, the series 6, 30, 150,... is a geometric sequence. The common ratio is 5, specifically. As a result, the formula for the sequence's nth term is:
[tex]a_n = a _1 * r^{(n-1) (n-1)}[/tex]
where a 1 denotes the first term, r the common ratio, and n the term number
[tex]a_n = 6 * 5^{(n-1) (n-1)}[/tex]
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Simplify n+1 combination 3 - n+1 combination 3
The simplified form of the combination is (3n² - 5n + 3)/3
Now let's return to the problem at hand. We have the expression ⁿ⁺¹C₃ − ⁿ⁻¹C₃, which involves two combinations with slightly different values of n. To simplify this expression, we need to first expand each combination using the formula we just discussed.
Starting with ⁿ⁺¹C₃, we have:
ⁿ⁺¹C₃ = (n + 1)!/(3!(n - (3 + 1))!)
We can simplify this further by noting that (n - 4)! is the same as (n - 3 - 1)! and pulling out the common factors:
(n + 1)!/(3!(n - 4)!) = (n + 1)(n)(n - 1)/(3 x 2 x 1)
Next, let's expand ⁿ⁻¹C₃:
ⁿ⁻¹C₃ = (n - 1)!/(3!(n - (3 - 1))!)
Again, we can simplify by noting that (n - 2)! is the same as (n - 1 - 1)!:
(n - 1)!/(3!(n - 2)!) = (n - 1)(n - 2)(n - 3)/(3 x 2 x 1)
Now we can substitute these expressions back into the original equation:
ⁿ⁺¹C₃ − ⁿ⁻¹C₃ = (n + 1)(n)(n - 1)/(3 x 2 x 1) - (n - 1)(n - 2)(n - 3)/(3 x 2 x 1)
We can simplify this by finding a common denominator and combining like terms:
[(n + 1)(n)(n - 1) - (n - 1)(n - 2)(n - 3)]/(3 x 2 x 1)
Simplifying further, we can expand the terms:
[(n³ - n) - (n³ - 7n² + 11n - 6)]/(6)
Which gives us:
(6n² - 10n + 6)/6
We can simplify this by factoring out a 2 from each term in the numerator and denominator:
2(3n² - 5n + 3)/2(3)
And finally, we can cancel out the 2s to get:
(3n² - 5n + 3)/3
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Complete Question:
Simplify:
ⁿ⁺¹C₃ − ⁿ⁻¹C₃
Lightbulbs act as resistors. Janine is building a circuit that contains two lightbulbs in parallel. One of the lightbulbs has a resistance of 120 ohms, but the resistance of the second lightbulb is unknown. She models the total resistance in the circuit, t, with this equation, in which r represents the resistance of the second lightbulb. T = 120 r r + 120 Find the inverse of Janine’s equation. Big fraction Parentheses Vertical bars Square root Root Superscript (Ctrl+Up) Subscript (Ctrl+Down) Plus sign Minus sign Middle dot Multiplication sign Equals sign Less-than sign Greater-than sign Less-than or equal to Greater-than or equal to Pi Alpha Beta Epsilon Theta Lambda Mu Rho Phi Sine Cosine Tangent Arcsine Arccosine Arctangent Cosecant Secant Cotangent Logarithm Logarithm to base n Natural logarithm Bar accent Right left arrow with under script Right arrow with under script Angle Triangle Parallel to Perpendicular Approximately equal to Tilde operator Degree sign Intersection Union Summation with under and over scripts Matrix with square brackets
Lightbulbs act as resistors. Janine is building a circuit that contains two lightbulbs in parallel. One of the lightbulbs has a resistance of 120 ohms, but the resistance of the second lightbulb is unknown. Therefore, The inverse of Janine's equation is r = -120/(t - 120).
Equation:
An equation is described as a formula that expresses the equality of two expressions, by connecting them with the equals sign. An equation is a formula that expresses that two expressions are equal by joining them with the equal sign =.The word comparison and its cognates in other languages may have subtly different meanings; for example, in French, an equation is defined to contain one or more variables, and in English any well-formed formula composed of two expressions linked by equal signs is an equation.
According to the Question:
In algebra, an equation is a mathematical statement that determines whether two mathematical expressions are equal.
The type of comparison is identity or conditional comparison.
For each potential value, the variable provides an identifier. Only certain combinations of variable values allow conditional comparisons to be true.
In order to find the inverse of Janine's equation, we will to solve for r in terms of t.
First, we can rearrange the equation to get
⇒ t - 120 = 1/r.
divide both sides of the equation by 1/r to get
⇒ t/r - 120/r = 1.
divide both sides of the equation by -120/r to get
⇒ r = -120/(t - 120).
Therefore, the inverse of Janine's equation is r = -120/(t - 120).
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By rounding to 1 significant figure , estimate the answer to the questions
216×876
The rounding of the number to 1 significant figure is-
216 × 876 = 180000
What is defined as the significant figure?The term significant figures describes the number of significant single digits (0 to several 9 inclusive) in a scientific notation coefficient.The number of significant figures inside an expression indicates the degree of certainty or precision with where an engineer or scientist states a number.All zeros to the right of the decimals but to the left of a non-zero number in a decimal number between 0 and 1 are not significant.0.00247, for example, only has three significant figures.216 × 876
This number can be written in form of rounding to 1 significant figure as;
200 × 900 = 180000
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Further Mathematics Assignment 1) A bowl contains 7 cooked and 5 uncooked In how many ways can: a) 2 cooked or 3 uncooked eggs be selected. b) 2 cooked and 3 uncooked eggs be selected.
Please this is my school homework!
Please HELP!!!
15 MRKS
Therefore, the total number of ways to select 2 cooked and 3 uncooked eggs is 210 ways.
What is combination?In mathematics, a combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. In other words, a combination is a selection of items from a larger set, without regard to the order in which they are chosen. The number of combinations of size r that can be chosen from a set of n elements is denoted by the symbol C(n, r), and is given by the formula:
C(n, r) = n! / (r! * (n-r)!)
where n! (pronounced "n factorial") is the product of all positive integers up to n, and r! is the product of all positive integers up to r.
Here,
a) To find the number of ways to select 2 cooked or 3 uncooked eggs, we need to use the combination formula, which is:
ⁿCₓ= n! / r!(n-r)!
where n is the total number of eggs, r is the number of eggs we want to select, and ! represents factorial.
For 2 cooked eggs, we can select them in:
⁷C₂ = 7! / 2!(7-2)! = 21 ways
For 3 uncooked eggs, we can select them in:
⁵C₃ = 5! / 3!(5-3)! = 10 ways
Therefore, the total number of ways to select 2 cooked or 3 uncooked eggs is:
21 + 10 = 31 ways
b) To find the number of ways to select 2 cooked and 3 uncooked eggs, we need to use the product rule of counting. That is, we need to multiply the number of ways to select 2 cooked eggs and 3 uncooked eggs.
The number of ways to select 2 cooked eggs is:
⁷C₂ = 21 ways
The number of ways to select 3 uncooked eggs is:⁵C₃ = 10 ways
Therefore, the total number of ways to select 2 cooked and 3 uncooked eggs is:
21 x 10 = 210 ways
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Write the first four terms of the sequence defined by a n = 5
{5, if n=1
a n -1 -5, if n>1
Answer:
The sequence is defined as follows:
a1 = 5
an = an-1 - 5, for n > 1
Using this definition, we can find the first four terms of the sequence as follows:
a1 = 5
a2 = a1 - 5 = 5 - 5 = 0
a3 = a2 - 5 = 0 - 5 = -5
a4 = a3 - 5 = -5 - 5 = -10
Therefore, the first four terms of the sequence are: 5, 0, -5, -10.
what is the surface area
Answer: 24
Step-by-step explanation:
Surface Area of a rectangular prism = 2 (lh +wh + lw ) Square units.
= 2[(2*2)+(2*2)+(2*2)]
= 2[4+4+4]
=2[12]
24
What is the y-intercept of the line with the equation y = 4 x + 2?
a. -2
b. 4
c. 2
d. -1/2
Answer:
C. 2-----------------------------------
The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0.
To find the y-intercept, substitute 0 for x in the equation y = 4x + 2:
y = 4*0 + 2 y = 2The y-intercept is 2 and matching choice is C.
Answer:
2
Step-by-step explanation:
We need to find out the y intercept of the given equation which is y = 4x + 2 .
y intercept is a point where the graph of the equation cuts the y axis. So at y intercept x coordinate will be 0 , so plug in x = 0 , in the given equation as ;
y = 4x + 2
y = 4(0) + 2
y = 0 + 2
y = 2
Therefore the y intercept of the line is 2 . ( option C )
the amount of bacteria present in a medium after t hours is given by a (t )equals 16 e to the power of 0.32 t end exponent. at what rate is the amount of bacteria changing after 12 hours?
The rate at which the amount of bacteria is changing after 12 hours is 21.698 units/hour.
What is the rate at which the amount of bacteria changing after 12 hours?The given formula for the amount of bacteria present in a medium after t hours is:
[tex]a(t) = 16e^ (0.32t)[/tex]
Now, we need to find out at what rate the amount of bacteria is changing after 12 hours.
This means we need to find out the derivative of a(t) with respect to t and then substitute t = 12 in the derivative formula to get the rate of change of bacteria after 12 hours.
Differentiating the given formula for a(t) with respect to t, we get:
a'(t) = [tex]16(0.32)e^(0.32t)[/tex]
On substituting, t = 12, we get
a'(12) = [tex]16(0.32)e^ (0.32X12)[/tex]
On solving this, we get a'(12) = 21.698.
Therefore, the rate at which the amount of bacteria is changing after 12 hours is 21.698 units/hour.
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