Answer: 729×64 is: (3^3 × 2^3)^2
Step-by-step explanation:
(a) To express 432 as the product of prime factors in exponential form, we can follow these steps:
Divide by 2 as many times as possible until the result is odd: 432 ÷ 2 = 216 ÷ 2 = 108 ÷ 2 = 54 ÷ 2 = 27 (5 times)
Divide by 3 as many times as possible until the result is not divisible by 3: 27 ÷ 3 = 9 ÷ 3 = 3 (2 times)
Since 3 is a prime number, we cannot divide by any other prime number to obtain a smaller result. Therefore, the prime factorization of 432 is: 2^4 × 3^3.
(b) To express 729×64 as the product of prime factors in exponential form, we can follow these steps:
Rewrite each factor as a power of a prime: 729 = 3^6 and 64 = 2^6.
Multiply the powers of each prime together: (3^6) × (2^6) = 3^6 × 2^6.
Simplify the result by factoring out the highest possible power of each prime: 3^6 × 2^6 = (3^3 × 2^3)^2.
Therefore, the prime factorization of 729×64 is: (3^3 × 2^3)^2.
Answer:
Below in bold.
Step-by-step explanation:
2) 432
2) 216
2) 108
2) 54
3) 27
3) 9
3
So 432 = 2^4 * 3^3.
3)729
3)243
3)81
3)27
3)9
3
64 = 2^6
So the answer is 2^6 * 3^6
Shade in the regions represented by the inequalities
Answer:
Step-by-step explanation:
see diagram
Which choice is an exponential function?
Of(x)=x²+6
Of(x)=5(3)*
Of(x) = 3x + 7
Of(x) = |2x+ 41
Answer:1st one
Step-by-step explanation:
17. The sum of the interior angles of a pentagon is6x + 6y. Find y in term of x
Answer:
We know that, the sum of interior angles of an n-sided polygon is (n−2)×180⁰
For a pentagon, n=5
so,
[tex] \implies \rm \: 6x + 6y = (n - 2)180[/tex]
[tex] \implies \rm \: 6(x + y) = (5 - 2)180[/tex]
[tex] \implies \rm \: 6(x + y )= 3 \times 180[/tex]
[tex] \implies \rm \: (x + y) = \dfrac{3 \times 180}{6} [/tex]
[tex] \implies \rm \: (x + y) = \dfrac{ 3 \times \cancel{180} \: \:30}{ \cancel6} [/tex]
[tex] \rm \implies \: x + y = 90[/tex]
[tex] \underline{\boxed{\implies \rm \: y= 90 - x}}[/tex]
Pentagon FormulasThere are many formulas related to a pentagon. A few basic ones are given below.
Diagonals of a pentagon: = n × (n − 3) ÷ 2 = 5 × (5 − 3) ÷ 2 = 5Sum of interior angles of a pentagon: = 180° × (n − 2) = 180° × (5 − 2) = 540°Each exterior angle of a regular pentagon: = 360° ÷ n = 360° ÷ 5 = 72°Each interior angle of regular pentagon: = 540° ÷ n = 540° ÷ 5 = 108°Area of a regular Pentagon = 1/2 × Perimeter × ApothemPerimeter of Pentagon = (side 1 + side 2 +side 3 + side 4 + side 5)Which of the following is equivalent to the inequality 2x + 13 < 5x - 20?
F. x >-11
G. x<?
H. x>;
J. x < 11
K. x > 11
Answer:
k
Step-by-step explanation:
2x+13<5x−20
Subtract 5x from both sides.
Combine 2x and −5x to get −3x.
Subtract 13 from both sides.
Subtract 13 from −20 to get −33.
Divide both sides by −3. Since −3 is negative, the inequality direction is changed.
x>11
Suppose a large candy machine has 45% orange candies. Imagine taking an SRS of 25 candies from the machine and observing the sample proportion
pof orange candies. Find the standard deviation of the sampling distribution of
pCheck to see if the 10% condition is met.
The standard deviation of the sampling distribution of pCheck to see if the 10% condition is 0.44.
If a large candy machine has 45% orange candies, and an SRS of 25 candies is taken from it, then the 10% condition can be checked to see if it is met. To do this, it is necessary to calculate the sample proportion p. To calculate p, the number of orange candies in the sample (let's call it x) is divided by the total number of candies in the sample (25). Thus, p = x/25. The 10% condition is met if the value of p is less than or equal to 0.1 (10%).
Since the value of p is unknown, the number of orange candies in the sample must first be determined. To find this number, the proportion of orange candies in the entire candy machine must be used. The proportion of orange candies is 45%, or 0.45.
This means that for every 100 candies in the candy machine, 45 are orange. Since there are 25 candies in the sample, 0.45*25 = 11.25 orange candies are expected in the sample. Since this is a sample proportion, the exact value of x must be rounded to the nearest whole number. This gives a value of 11 orange candies in the sample.
Now that the value of x is known, the value of p can be calculated. This gives p = 11/25 = 0.44.
Since 0.44 is greater than 0.1 (10%), the 10% condition is not met in this case.
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A plan for a house is drawn on a 1:40 scale. If the length of the living room on the plan measures 4.5 inches, what is the actual length of the built living room? 45 feet 25 feet 15 feet 12 feet
Answer:
actual length = 15 feet
Step-by-step explanation:
using the conversion
12 inches = 1 foot
the actual length = 40 × scale length = 40 × 4.5 = 180 inches = 180 ÷ 12 = 15 feet
Write a formula for f(t) as a sum of Heaviside functions. Type uc for the Heaviside function that jumps at c (don't type uc(t) ). F(t) = t, 0 ≤ t ≤ 3 2t−3, 3 < t ≤ 4 5, 4 < t
The formula for f(t) as a sum of Heaviside functions is f(t) = tuc(t) - tuc(t-3) + (2t-3)uc(t-3) - (2t-3)uc(t-4) + 5uc(t-4).
To express f(t) as a sum of Heaviside functions, we break down the function into its respective intervals and use the unit step function or Heaviside function to represent the function in each interval.
f(t) = t, 0 ≤ t ≤ 3
f(t) = 2t - 3, 3 < t ≤ 4
f(t) = 5, 4 < t
Using the Heaviside function uc for the jump at c, we can represent each interval as follows:
f(t) = tuc(t) - tuc(t-3), 0 ≤ t ≤ 3
f(t) = (2t-3)uc(t-3) - (2t-3)uc(t-4), 3 < t ≤ 4
f(t) = 5uc(t-4), 4 < t
Therefore, the formula is f(t) = tuc(t) - tuc(t-3) + (2t-3)uc(t-3) - (2t-3)uc(t-4) + 5uc(t-4)
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is every point of every open set e c r2 a limit point of e? answer the same question for closed sets in r2
All points belonging to any closed set in R2 are limit points of that set.
What is a limit point?A limit point is a point in a metric space that can be approached arbitrarily closely by points of a sequence or a set. The definition is different from that of an isolated point, which has an open ball around it that does not include any other points of the set.
According to the definition of a limit point, every point in every open set in R2 is a limit point of the set. This is because there is always at least one other point in the open set that is within a specified distance of the point in question.
As a result, every point in every open set in R2 is a limit point of the set. We can answer the same question for closed sets in R2. A closed set in R2 contains all of its limit points. A closed set, by definition, includes its boundary.
As a result, every point in every closed set in R2 is a limit point of the set.
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Solve the system of equations
n=2d+5
0. 05n+0. 10d=2. 25
Answer: THIS IS THE ANSWER
HELP. I'm really struggling on this one. My calculus teacher claimed this to be the easiest math problem ever but I still can't understand. Is anyone smart enough to figure this one out. Whats 1 + 1?
Answer:
The answer to 1 + 1 is 2.
Very complicated problem, please mark brainliest!
Answer:
1+1 = 2
Or, 1=2-1
1=1
we know value of one is one
so,
1+1=11
use a blank number line to solve. which of the following expressions have a value that is less than 0? select all that apply. Sorry is it says i am in high school i am in 7th grade
a)-5+3
b)6+(-8)
c)5+(-3)
d)-2+5
The expressions that have values less than 0 are a) and b)
According to the question
Here's how you can use a number line to solve this problem:
Draw a blank number line with a zero in the middle.For each expression, start at zero and move to the right or left depending on the sign of the first number.Then, add or subtract the second number to get the final position on the number line.If the final position is to the left of zero, the expression has a value that is less than 0.For example in expression a) -5 + 3, start at zero and move 5 units to the left (because of the negative sign on 5), and then move 3 units to the right. The final position is at -2, which is to the left of zero. Therefore, expression a) has a value that is less than zero.
Similarly expression b) has a value that is less than zero.
Also expressions c) and d) can be solved in the same way. If the final position is to the right of zero, the expression does not have a value that is less than zero.
In conclusion, expressions a) and b) have a value that is less than zero, while expressions c) and d) do not.
What is a number line?
A number line is a visual representation of real numbers arranged in a straight line used to compare, add, and subtract numbers.
What is meant by expressions?
An expression is a combination of numbers, variables, and mathematical operations used to represent a quantity or a mathematical statement.
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Subtract (34ab – 15b + 8a) from (20ab +16b +4a)
Answer:
-14ab + 31b -4a
Step-by-step explanation:
(20ab + 16 b + 4a) - (34ab - 15b +8a)
20 ab +16b + 4a -34ab + 15b -8a
-14ab + 31b -4a
The table contains data on the number of people visiting a historical landmark over a period of one week.
Which type of function best models the relationship between the day and the number of visitors?
A. A quadratic function with a negative value of a.
B. A quadratic function with a positive value of a.
C. A square root function.
D. A linear function with a positive slope.
Hence , it is to be quadratic function where a has to be less than zero or
A quadratic function with a negative value of a.
Given that ,
The table contains data on the number of people visiting a historical landmark over a period of one week.
We have to find,
The relationship between the day and the number of visitors.
According to the question,
y = a√x +b
And where x = no. of days = 1
y = no. of visitors = 45
45 = a √1 + b
45 = a + b
And When x = 2 and y = 86
86 = a √2 +b
Solving the equation for the values of a and b.
From equation 1
45 - a = b
Put the value of b in equation 2
⇒ 86 = + 45 - a
⇒ 86 - 45 = a √2 - a
⇒ 41 = a (√2 - 1 )
⇒ a = [tex]\frac{41}{\sqrt{2}- 1 }[/tex]
⇒ a = 41/0.41
⇒ a = 100
Put the value of a = 100
45 = 100 + b
45 - 100 = b
b = -55
Now,
The required equation is y = 100 -55.
When x = 4
y = 100√2 - 55
⇒ y = 100(1.4) - 55
⇒ y = 140 - 55
⇒ y = 85
And,
when x = 5
y = 100√5 - 55
⇒ y = 223.5 - 55
⇒ y = 168.60
⇒ y = 168 ( approx. )
Therefore, 168 ≠ 158
Since the function rises less and less in later stage , it can not be a first order function.
So, it is to be quadratic function where a has to be less than zero.
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please help I have attached a photo below. thanks for your time and help.
According to the given image, with the help of a protractor, we can conclude that ∠NCD represents an angle of 50°.
What is a protractor?An instrument for measuring angles is a protractor, which is often made of transparent plastic or glass.
Protractors might be straightforward half-discs or complete circles. Protractors with more complex features, like the bevel protractor, include one or two swinging arms that can be used to measure angles.
Most protractors use degrees (°) to express angles.
Protractors with radian scales calculate angles.
The majority of protractors have 180 equal segments.
Degrees are further subdivided into arcminutes by some precision protractors.
They are employed in geometry, engineering, and mechanics.
So, in the given image the proctor is measuring the angle of 50°.
Therefore, according to the given image, with the help of a protractor, we can conclude that ∠NCD represents an angle of 50°.
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Each of these measures is rounded to nearest whole: a=5cm and b=3cm Calculate the upper bound of a +b
The upper bound of a + b can be found by adding the upper bounds of a and b.
For a = 5cm, the nearest whole number is 5. The upper bound would be the midpoint between 5 and 6, which is 5.5.
For b = 3cm, the nearest whole number is 3. The upper bound would be the midpoint between 3 and 4, which is 3.5.
So the upper bound of a + b is:
5.5 + 3.5 = 9
Therefore, the upper bound of a + b is 9cm.
The Jones family has two dogs whose ages add up to 15 and multiply to 44. How old is each dog?
Hi!
Your answer is 11 and 4.
11 x 4 = 44
11 + 4 = 15
Hope this helps!
~~~PicklePoppers~~~
Use the image below to get the measure of ∠w
Answer:49
Step-by-step explanation:
The planet XYZ traveling about the star ABC in a circular orbit takes 24 hours to make an orbit. Assume that the orbit is a circle with radius 83,000,000 mi. Find the linear speed of XYZ in miles per hour. The linear speed is approximately ______ miles per hour. (Round to the nearest integer as needed.)
The planet XYZ traveling about the star ABC in a circular orbit takes 24 hours to make an orbit. Assume that the orbit is a circle with a radius of 83,000,000 mi. The linear speed of XYZ in miles per hour is approximately 1,093,333 miles per hour.
What is the orbit and what is the linear speed?
An orbit refers to the path taken by an object, such as a planet, as it circles around another object, such as a star. The speed of the planet is its rate of movement, measured in linear units like miles or kilometers per hour, as it travels around the orbit.
These are terms that are important to understanding the solution to the problem provided. The linear speed of XYZ in miles per hour is approximate _____ miles per hour. (Round to the nearest integer as needed.)
The planet XYZ travels around the star ABC in a circular orbit that takes 24 hours to complete. The orbit is a circle with a radius of 83,000,000 miles.
To find the linear speed of XYZ in miles per hour, it is necessary to use the formula for the circumference of a circle.
Circumference = 2πr Circumference
=2πr Substitute 83,000,000 for r in the formula.
Circumference = 2π(83,000,000)
Circumference = 522,000,000 π
The orbit's circumference is 522,000,000 π miles.
The distance traveled by XYZ in one hour is the linear speed. The linear speed of XYZ in miles per hour is calculated as follows:
Speed = Distance/TimeSpeed
= Circumference/24Speed
= (522,000,000 π)/24
Speed = 21,750,000 π
The linear speed of XYZ in miles per hour is 21,750,000 π miles per hour.
To get an approximate answer, π is equal to 3.14.
Speed ≈ 21,750,000 (3.14)
Speed ≈ 68,295,000
The linear speed of XYZ in miles per hour is approximately 68,295,000 miles per hour. Rounded to the nearest integer, the linear speed is approximately 1,093,333 miles per hour.
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Solve for x. Simplify your answer.
Therefore, the length of the larger part is 9 units.
What is ratio?A ratio is a comparison of two quantities that measures how many times one quantity is contained within the other. It is expressed using the symbol ":" or as a fraction. For example, a ratio of 2:3 (or 2/3) means that one quantity is two-thirds the size of the other quantity. Ratios can be used to compare various quantities, such as length, area, volume, weight, or even numbers of objects. They are commonly used in mathematics, finance, and many other fields.
Here,
If the line segments are divided in the ratio of 2:3, this means that the smaller part is 2 units and the larger part is 3 units. We know that the length of the smaller part is 6 units, so we can set up a proportion:
2/3 = 6/x
where x is the length of the larger part. We can solve for x by cross-multiplying:
2x = 3*6
2x = 18
x = 9
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Kevin made a down payment of $1,500 for his car lease. His monthly payment amount is $200 for 36 months. What is the total cost of the lease if Kevin purchases the car. The residual value at the end of the lease is $8,000?
Kevin made a down payment of $1500 for his automobile leasing. Kevin's monthly payment for a 36-month rental with an overall cost of $4,200 will be $200 if he opts to buy the vehicle outright for $16,700.
What kind of down payment is this, exactly?A deposit on a property is a typical illustration of a down payment. The down payment for a property can range from 5% to 25%, with the remaining amount being covered by a mortgage obtained from a bank or another financial institution.
A down payment or a deposit is it?A commitment is retained by a 3rd person in trust, only until date of completion when it constitutes a component of the deposit.
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The complete question is:
Kevin made a down payment of $1,500 for his car lease. His monthly payment amount is $200 for 36 months. What is the total cost of the lease if Kevin purchases the car. The residual value at the end of the lease is $8,000?
a. $14,500
b. $15,200
c. $16,700
Answer: 16,700
Step-by-step explanation:
....
Simplify these radicals (ASAP!!!)
Answer:
[tex]3.\quad 2\sqrt{7}\\\\4.\quad \dfrac{2}{\sqrt{5}}[/tex]
[tex]5.\quad \dfrac{1}{\sqrt{2}}[/tex]
[tex]6.\quad\dfrac{5}{7}[/tex]
Step-by-step explanation:
To simplify radicals find the factors and simplify the square roots of the factors
#3 : √28
28 = 4 x 7
√28 = √(4 x 7) = √4 x √7 = 2 x √7 = 2√7
========================================================
[tex]\#4:\quad \sqrt{ \dfrac{4}{5}}\\\\\sqrt{ \dfrac{4}{5}} = \dfrac{\sqrt{4}}{\sqrt{5}} = \dfrac{2}{\sqrt{5}}[/tex]
========================================================
[tex]\#5\quad \dfrac{3}{\sqrt{18}}\\\text{Denominator } \sqrt{18 }= \sqrt{9 \times 2} = 3\sqrt{2}\\\\ \dfrac{3}{\sqrt{18}}= \dfrac{3}{3\sqrt{2}} = \dfrac{1}{\sqrt{2}}[/tex]
========================================================
[tex]\# 6 \quad \sqrt{\dfrac{25}{49}}\\\\\sqrt{25} = 5\\\\\\\sqrt{49} = 7\\\\[/tex]
[tex]\sqrt{\dfrac{25}{49}}= \dfrac{5}{7}[/tex]
11. How much time will it take for ₹5000
5618 at 6% per annum
annually?
to become
compounded
Answer:
2.31 Years
Step-by-step explanation:
To calculate the time it will take for ₹5000 to grow to ₹5618 with a 6% annual interest rate when compounded annually, we can use the following formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (₹5618)
P = the principal amount (₹5000)
r = the annual interest rate (6% or 0.06)
n = the number of times the interest is compounded per year (1, since it's compounded annually)
t = the time period in years
Plugging in the values, we get:
5618 = 5000(1 + 0.06/1)^(1t)
Simplifying:
1.1236 = 1.06^t
Taking the natural logarithm of both sides:
ln(1.1236) = ln(1.06^t)
Using the power rule of logarithms:
ln(1.1236) = t ln(1.06)
Solving for t:
t = ln(1.1236) / ln(1.06)
t ≈ 2.31 years
Therefore, it will take approximately 2.31 years for ₹5000 to grow to ₹5618 at a 6% annual interest rate when compounded annually.
Find the volume of the washer. Round to the nearest whole millimeter
The volume of the washer as per the given dimensions is calculated to be 7,854 mm³.
In order to calculate the volume of the washer, we are required to subtract the volume of the hole in the center from the volume of the solid cylinder with the outer radius. The volume of the solid cylinder with the outer radius can be calculated using the formula:
V1 = πr1²h (Here, r1 is the outer radius and h is the thickness)
Substituting the given values, we have:
V1 = π(30mm)²(5mm) = 14,137.17 mm³
The volume of the hole in the center can be calculated using the same formula, but with the inner radius,
V2 = πr2²h (Here r2 is the inner radius and h is the thickness)
Substituting the given values, we have:
V2 = π(20mm)²(5mm) = 6,283.19 mm³
Therefore, the volume of the washer is:
V = V1 - V2 = 14,137.17 mm³ - 6,283.19 mm³ = 7,853.98 mm³
Therefore, on rounding the obtained result to the nearest whole millimeter, the final volume is calculated to be 7,854 mm³.
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The complete question is :
What is the volume of a washer with an outer radius of 30 mm, an inner radius of 20 mm, and a thickness of 5 mm? (Round your answer to the nearest whole millimeter)
Formula: m=;
X₂-X1
P₁ = (X₁, Y₁) P₂ = (X2, Y₂)
1) P₁=(-1,5) y P₂=(-2,-6)
Cómo resolver este ejercicio de ecuaciones lineales de dos puntos
The equation m = 12 + 2Y1 gives the slope of the line through the points P1=(-1.5, Y1) and P2=(-2, -6)
In mathematics, how do you find coordinates?To find a point's coordinates in a coordinate system, you perform the opposite. Starting at the point, move vertically up or down until you reach the x-axis. Your x-coordinate is shown there. Afterwards, you'll be able to get a better understanding of how to use it.
It is assumed that the formula you supplied would be used to determine the slope (m) of the line connecting the two specified points, P1 and P2.
We may use the following formula to find the slope (m) of the line passing through the coordinates P1=(X1, Y1) and P2=(X2, Y2):m = (Y2 - Y1) / (X2 - X1) (X2 - X1)
Using the values provided:
P1 = (-1.5, Y1) (-1.5, Y1)
P2 = (-2, -6) (-2, -6)
These values can be used as substitutes in the formula:
m = (-6 - Y1) / (-2 - (-1.5))
Making the denominator simpler:
m = (-6 - Y1) / (-2 + 1.5)
m = (-6 - Y1) / (-0.5)
m = 12 + 2Y1
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Question:
Formula: m=;
X₂-X1
P₁ = (X₁, Y₁) P₂ = (X2, Y₂)
1) P₁=(-1.5) and P₂=(-2,-6)
How to solve this exercise of linear equations of two points.
Write as a single power of 3:
27divided by 9a
Answer:
Step-by-step explanation:
27/9a
= 3^3/3^2 a
= 3/a
An experiment consists of rolling two fair number cubes. The diagram shows the sample space of all equally likely outcomes. What is the probability of rolling two 1's? Express your answer as a fraction in simplest form.
The probability of rolling two 1's is 1/18. This concludes the answer.
Why it is and what does probability mean?
Probability = No. of favorable outcomes/Total Outcomes
Here the no of favorable outcomes = 2{(2,1), (1,2) }
Total outcomes = 36
P (1 and 2) = 2/36 = 1/18
Therefore, the solution is 1/18
Probability is a branch of mathematics that deals with the study of random events or outcomes. It is used to quantify the likelihood of an event occurring, which ranges from 0 (impossible) to 1 (certain). Probability is expressed as a ratio, fraction, or percentage. The concept of probability is used in various fields such as science, economics, finance, and statistics to make informed decisions and predictions.
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104.
Simplify the polynomial by writing each of its term in standard form.
12a^2 · 3ba − 2ab · 3ab^2+11aba
The simplified polynomial, with each term written in standard form, is: 36a^3b - 6a^2b^3 + 11a^2b
How to simplify the polynomialTo simplify the given polynomial, we need to expand and combine like terms.
First, we can distribute the coefficients of the first term:
12a^2 · 3ba = 36a^3b
Next, we can simplify the second term by multiplying the coefficients and adding the exponents:
-2ab · 3ab^2 = -6a^2b^3
Finally, we can combine the like terms:
36a^3b - 6a^2b^3 + 11a^2b
To write each term in standard form, we arrange the terms in decreasing order of exponents of 'a' and 'b':
36a^3b - 6a^2b^3 + 11a^2b
So, the simplified polynomial, with each term written in standard form, is:
36a^3b - 6a^2b^3 + 11a^2b
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NEED THIS ANSWERED ASAP!!
The line of site to the horizon would be tangent to the Earth’s surface. What kind of angle is formed between the radius of the Earth and the line of site?
Answer:
right angle
Step-by-step explanation:
You want to know the kind of angle formed between a radius and a tangent.
TangentA tangent to a circle is always perpendicular to the radius at the point of tangency.
The angle is a right angle.
Past studies indicate that about 60 percent of the trees in a forested region are classified assoftwood. A botanist studying the region suspects that the proportion might be greater than 0.60.The botanist obtained a random sample of trees from the region and conducted a test ofB:p=0.6 versus H, :p>0.6. The P-value of the test was 0.015. Which of the following is acorrect interpretation of the P-value?a. if it is true that 60 percent of the trees in a forested region are classified as softwood, 0.015 is theprobability of obtaining a population proportion greater than 0.6.b. If it is true that 60 percent of the trees in a forested region are classified as softwood, 0.015 is the8 probability of obtaining a sample proportion as small as or smaller than the one obtained by thebotanistc. If it is true that 60 percent of the trees in a forested region are classified as softwood, 0.015 is theprobability of obtaining a sample proportion as large as or larger than the one obtained by thebotanistd. If it is not true that 60 percent of the trees in a forested region are classified as softwood, 0.015 isthe probability of obtaining a sample proportion as large as or larger than the one obtained by thebotaniste. If it is not true that 60 percent of the trees in a forested region are classified as softwood, 0.015 isthe probability of obtaining a population proportion greater than 0.6.
If it is true that 60 percent of the trees in a forested region are classified as softwood, 0.015 is the probability of obtaining a sample proportion as large as or larger than the one obtained by the botanist.The correct answer is option C
The P-value is the probability of obtaining an outcome as extreme or more extreme than the one actually observed given that the null hypothesis is true. In this case, the null hypothesis is that the proportion of trees classified as softwood is 0.60. The P-value of 0.015 indicates that the probability of obtaining a sample proportion as large as or larger than the one obtained by the botanist is 0.015 if the null hypothesis is true.
Therefore, the P-value is an indicator of how well the sample data supports the null hypothesis. In this case, the P-value of 0.015 is lower than the commonly accepted significance level of 0.05, which indicates that the sample data does not support the null hypothesis and suggests that the proportion of trees classified as softwood is actually greater than 0.60.
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2. Oil is leaking from an uncapped well and polluting a lake. Ten days after the leak is discovered, environmental engineers measure the amount of oil in the water to be 200 gallons with a current inflow rate of 30 gallons per day. The leak is slowing so that on the tenth day, the inflow rate is decreasing by 5 gallons/day each day. Suppose Q(t) is the amount of oil (in gallons) t days after the leak is discovered. (a) Find the second degree Taylor polynomial for Q(t) centered at t=10. (b) Use your answer in the previous part to estimate the amount of oil in the lake at t=12
As a result, 210 gallons of oil are thought to have been present in the lake at time 12 (t=12).
what is polynomial ?A polynomial is a mathematical equation that only uses addition, subtraction, multiplication, and non-negative integer exponents and is made up of variables and coefficients. To put it another way, a polynomial is an algebraic expression made up of terms that are sums, products, and/or products of variables and coefficients. The leading coefficient is the coefficient of the term with the highest degree, while the degree of a polynomial is the maximum power of the variable in the expression.
given
(a) We need to determine Q(10), Q'(10), and Q" in order to determine the second degree Taylor polynomial for Q(t) with a center at t=10 (10).
As we are aware, Q(10) = 200. (given in the problem statement).
We must take the derivative of Q(t) with respect to t in order to determine Q'(10):
Q'(t) = -5t + 250
Q'(10) = -5(10) + 250 = 200
We must take the derivative of Q'(t) with respect to t in order to determine Q"(10):
Q''(t) = -5
Q''(10) = -5
The second degree Taylor polynomial can now be calculated using the following formula: P2(t) = Q(10) + Q'(10)(t-10) + (1/2)Q"(10)(t-10)2.
With the numbers we discovered, we can calculate P2(t) as 200 + 200(t-10) + (1/2)(-5)(t-10)2.
[tex]P2(t) = 200 + 200(t-10) - (5/2)(t-10)^2[/tex]
(b) We must assess the second degree Taylor polynomial at t=12 in order to determine how much oil is in the lake at that time.
P2(12) = 210 gallons because P2(12) = 200 + 200(12-10) - (5/2)(12-10)2. (rounded to the nearest gallon)
As a result, 210 gallons of oil are thought to have been present in the lake at time 12 (t=12).
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