To solve the exact differential equation (5t^2 + 8y) dy + (10yt + 9t^2) = 0, we need to check if it is exact or not. We do so by taking partial derivatives with respect to y and t:
∂/∂y (5t^2 + 8y) = 8
∂/∂t (10yt + 9t^2) = 10y + 18t
Since these partial derivatives are not equal, the equation is not exact. To make it exact, we can multiply the entire equation by a integrating factor, which is given by:
μ = e^(∫(∂/∂t)(10yt + 9t^2) dt) = e^(∫(10y + 18t) dt) = e^(10yt + 9t^2)
Multiplying both sides of the equation by μ, we get:
(5t^2 + 8y)e^(10yt + 9t^2) dy + (10yt + 9t^2)e^(10yt + 9t^2) dt = 0
Now, we can check if this equation is exact:
∂/∂y (5t^2e^(10yt + 9t^2) + 8ye^(10yt + 9t^2)) = 10te^(10yt + 9t^2)
∂/∂t ((10ye^(10yt + 9t^2)) + (9t^2e^(10yt + 9t^2))) = 10ye^(10yt + 9t^2) + 18t^2e^(10yt + 9t^2)
These partial derivatives are equal, so the equation is exact. Therefore, we can find a potential function Φ such that:
∂Φ/∂y = 5t^2e^(10yt + 9t^2) + 8ye^(10yt + 9t^2)
∂Φ/∂t = (10ye^(10yt + 9t^2)) + (9t^2e^(10yt + 9t^2))
Integrating the first equation with respect to y, we get:
Φ = ∫(5t^2e^(10yt + 9t^2) + 8ye^(10yt + 9t^2)) dy = (5t^2/10)e^(10yt + 9t^2) + (4y/10)e^(10yt + 9t^2) + C(t)
where C(t) is an arbitrary constant of integration that depends only on t.
Now, we can differentiate this expression with respect to t and compare it to the second equation:
∂Φ/∂t = (10t/10)e^(10yt + 9t^2) + C'(t)
(10ye^(10yt + 9t^2)) + (9t^2e^(10yt + 9t^2)) = (10t/10)e^(10yt + 9t^2) + C'(t)
Comparing the two expressions, we get:
C'(t) = 10ye^(10yt + 9t^2)
Integrating both sides with respect to t, we get:
C(t) = ∫10ye^(10yt + 9t^2) dt = e^(10yt + 9t^2) + K
where K is another arbitrary constant of integration.
Therefore, the solution to the exact differential equation (5t^2 + 8y) dy + (10yt + 9t^2) = 0 is given by:
(5t^2/10)e^(10yt + 9t^2) + (4y/10)e^(10yt + 9t^2) + e^(10yt + 9t^2) + K = 0
or simplifying:
y = (-5t^2/4) - (1/2)e^(-10yt - 9t^2) - (K/4)e^(-10yt - 9t^2)
where K is an arbitrary constant of integration.
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determine the ordered pairs of....
[tex]6x - y > - 3[/tex]
and
[tex]4x + 3y < 4[/tex]
The ordered pairs of the system of inequalities are (1, 0) and (0, -5)
Determining the ordered pairs of the system of inequalitiesFrom the question, we have the following parameters that can be used in our computation:
6x - y > -3
4x + 3y < 4
The above expression is a system of linear inequality
That implies that we graph the inequalites in the system on the same plane and write out ordered pairs from the region that represent the solution of the system
Next, we plot the graph
See attachment for the graph of the inequality
The ordered pairs are (1, 0), (0, -5) and other pairs
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The data set is 12, 46, 32, 18, 26, 41, 46. the mean is 31.6 and the median is 32. if we add another 12, what affect does this have on the mean and median?
Adding another 12 to the data set would increase the sum of the values by 12, resulting in a new sum of 239. To find the new mean, we divide the new sum by the total number of values in the set, which is now 8. So the new mean would be 29.875, which is slightly lower than the original mean of 31.6.
To find the new median, we first need to rearrange the values in ascending order: 12, 18, 26, 32, 41, 46, 46, 12. Since there are now an even number of values, we take the average of the middle two, which in this case is (26 + 32) / 2 = 29. So the new median would be 29, which is lower than the original median of 32.
In summary, adding another 12 to the data set would slightly decrease the mean and lower the median.
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The Integral ∫55dx/√86x - x^2 can converges to
The integral ∫(5/5)dx/√[86(x^2 - x^2)] converges to 5 since the denominator becomes 0 at x=0, which is not in the interval of integration [5,5].
We can start by simplifying the integrand
∫(5/5)dx/√[86(x^2 - x^2)]
Using the identity a^2 - b^2 = (a + b)(a - b), we can rewrite the denominator as
√[86(x^2 - x^2)] = √[86(x + x)(x - x)] = √[86] * √[x + x] * √[x - x] = √[86] * √[2x] * √[0] = 0
Therefore, the integrand is undefined when x = 0. Since the interval of integration is [5,5], which does not include 0, the integral is well-defined and converges to 5.
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If you vertically compress the absolute value parent function, f(x) = x1, by a
factor of 4, what is the equation of the new function?
O A. G(x) = (x-41
B. G(x) = 1
O C. G(x) = 14x1
O (
D. G(x) = 411
Equation of new function when vertically compressing the absolute value of parent function by a factor of 4 is option d. g(x) = (1/4)|x|.
The absolute value parent function is f(x) = |x| .
f(x) = x when x is positive,
and f(x) = -x when x is negative.
To vertically compress the function by a factor of 4,
Multiply the function by 1/4.
This implies,
The equation of the new function is equal to,
g(x) = (1/4) × f(x)
= (1/4) × |x|
= (1/4) × x when x is positive,
and g(x) = (1/4) × (-x)
= (-1/4) × x when x is negative.
This implies,
g(x)= (1/4) × f(x)
= (1/4) |x|
(1/4) x for x ≥ 0
(-1/4) x for x < 0
Therefore, the equation of the new function is equal to option d. g(x) = (1/4)|x|.
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The above question is incomplete, the complete question is:
If you vertically compress the absolute value parent function, f(x) = |x|, by a factor of 4, what is the equation of the new function?
a. g(x) = 4x
b. g(x) = 4x -1
c. g(x) = x - 4
d. g(x) = (1/4)|x|
Solve for x: √8x + 4 = 6
The solution to the equation √8x + 4 = 6 is x = 0.5.
What is the value of x?An equation is simply a mathematical formula that expresses the equality of two expressions, using the equals sign as a connection between them.
Given the equation in the question:
√8x + 4 = 6
To solve for x in the equation, isolate the term containing the variable x.
Subtract 4 from both sides of the equation:
√8x + 4 - 4 = 6 - 4
√8x = 6 - 4
√8x = 2
Square both sides of the equation:
( √8x )² = 2²
8x = 4
Divide both sides of the equation by 8:
x = 4/8
x = 1/2
x = 0.5
Therefore, the value of x is 0.5.
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Target INT1: I can correctly antidifferentiate basic functions and identify antiderivatives nd the most general antiderivatives of each function. 1. y = 1 - x^2 + x^2 + 3x^4
2. g(x) = cos x + x
3. h(x) = 5
The antiderivative of h(x) is 5x + C.
We can simplify the function y as y = 1 + 3x^4. Now, we can integrate term by term as:
∫ y dx = ∫ (1 + 3x^4) dx
= x + (3/5)x^5 + C
So, the antiderivative of y is x + (3/5)x^5 + C.
The antiderivative of cos(x) is sin(x) and the antiderivative of x is (1/2)x^2. Therefore, we can integrate term by term as:
∫ g(x) dx = ∫ (cos x + x) dx
= sin(x) + (1/2)x^2 + C
So, the antiderivative of g(x) is sin(x) + (1/2)x^2 + C.
The antiderivative of any constant is the constant times x. Therefore, we can integrate h(x) as:
∫ h(x) dx = ∫ 5 dx
= 5x + C
So, the antiderivative of h(x) is 5x + C.
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Jose conducted an experiment to measure the rate of minerals dissolving in water and changed the temperature of the water for each trial.
What is the independent variable in this experiment?
A: number of trials being tested
B: temperature of the water
C: type of minerals used for each trial
D: rate the minerals dissolved
The independent variable in the study is the temperature of the water
What is the independent variable?The variable that the experimenter consciously modifies or changes is known as the independent variable.
The variable that is supposed to have an impact on the dependent variable is this one. After then, the dependent variable is assessed and recorded to see if it changes as a result of the independent variable's manipulation.
The temperature was changed before each trial hence the temperature was manipulated and that is the independent variable.
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A researcher is studying life expectancy in different parts of the world using birth and death records she randomly selects a sample of 20 people from town a and a samle of 20 people form town b and records their lifespans in years.
mean lifespan in years standard deviation
town a 78.5 11.2
town b 74.4 12.3
the researhers wants to test the claim that there is a significant differnce in lifepan for people in the two towns. what are the null and alternative hypotheses that should be used to test this claim?
a. null: mu1 - mu2 is not equal to 0; alternative: mu1 - mu2 > 0 <---- a
b. null: mu1 - mu2 = 0; alternative: p1 - p2 is not equal to 0 <---- b
c. null: mu1 - mu2 = 0; alternative: mu1 - mu2 is not equal to 0 <---- c
d. null: p1 - p2 = 0; alternative: p1 - p2 < 0 <---- d
The correct answer is: c. null: mu1 - mu2 = 0; alternative: mu1 - mu2 is not equal to 0
The null hypothesis states that there is no significant difference in the mean lifespan of people in the two towns, while the alternative hypothesis states that there is a significant difference in the mean lifespan of people in the two towns.
Since we are comparing the means of two populations, we use the difference between the means as the test statistic. Therefore, the null hypothesis is mu1 - mu2 = 0 (there is no difference between the means) and the alternative hypothesis is mu1 - mu2 is not equal to 0 (there is a difference between the means).
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Create a table of values for the function y=x² - 4x + 12 for the domain {-2,-1,0,1}. What is the value of f¹(12)?
Answer:
f¹(12) = 20
Step-by-step explanation:
f(-2) = 24
f(-1) = 17
f(0) = 12
f(1) = 9
f¹(x) = 2x - 4
f¹(12) = 20
You are going to make a password that starts with two letters from the alphabet, followed by three digits (for example, AB-123). Digits may be numbers 0
through 9
If you are allowed to repeat letters or numbers, you can make
passwords.
If you don't repeat any letters or numbers, you can make
passwords
When allowing repetition, you can make 676,000 passwords, and without repetition, you can make 468,000 passwords.
To create a password that starts with two letters from the alphabet, followed by three digits (for example, AB-123), you can make a different number of passwords depending on whether you are allowed to repeat letters or numbers.
1. If you are allowed to repeat letters or numbers, you can make:
- 26 (alphabet letters) x 26 (alphabet letters) x 10 (digits 0-9) x 10 (digits 0-9) x 10 (digits 0-9) = 676,000 passwords.
2. If you don't repeat any letters or numbers, you can make:
- 26 (alphabet letters) x 25 (remaining alphabet letters) x 10 (digits 0-9) x 9 (remaining digits 0-9) x 8 (remaining digits 0-9) = 468,000 passwords.
Your answer: When allowing repetition, you can make 676,000 passwords, and without repetition, you can make 468,000 passwords.
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In a survey, the planning value for the population proportion is p* = 0.25. How large a sample should be taken to provide a 95% confidence interval with a margin of error of 0.02? Round your answer up to the next whole number. How large a sample should be selected to provide a 95% confidence interval with a margin of error of 2? Assume that the population standard deviation is 30. Round your answer to next whole number.
The Sample size that is necessary for the selection is =7203
How to solveGiven that,
[tex]\hat p= 0.25[/tex]
[tex]1 - \hat p = 1 - 0.25 = 0.75[/tex]
margin of error = E = 0.01
At 95% confidence level the z is ,
\alpha = 1 - 95% = 1 - 0.95 = 0.05
[tex]\alpha / 2 = 0.05 / 2 = 0.025[/tex]
Z\alpha/2 = Z0.025 = 1.96 ( Using z table )
Sample size = n = [tex](Z\alpha/2 / E)2 * \hat p * (1 - \hat p)[/tex]
= (1.96 / 0.01)2 * 0.25 * 0.75
= 7203
Sample size =7203
In statistics, the sample size is the measure of the number of individual samples used in an experiment.
The size of the sample holds significant importance in any empirical study that aims to draw conclusions about a larger population based on a smaller sample.
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When estimating population parameters, a point estimate is: group of answer choices the population mean a statistic that estimates a population parameter a range of possible values for a population parameter always equal to a population value
When estimating population parameters, a point estimate is: a population parameter
What is a point estimatePoint estimates are statistical estimates used to approximate population parameters such as mean, proportion or variance that remain unknown.
They provide one value as an approximation for unknown parameters in a population sample that may or may not match up exactly with true population value; nevertheless they serve as reasonable approximations.
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HELPPPPP FAST PLEASEEEE
Answer:
AEC is similar to BDC in terms of the type of triangle
That's all I can really say
I hope this helps :)
How much Pure alcohol must a pharamacist add to 10cm cubed of a 8% alcohol solution to strengthen it to a 80% solution
The amount of Pure alcohol must a pharamacist add to 10cm cubed of a 8% alcohol solution to strengthen it to a 80% solution is 36 cm³.
Alcohol, also known as ethanol is a clear, colorless liquid that is produced by the fermentation of sugars and carbohydrates by yeasts.
Let's start by writing down the equation that relates the amount of alcohol in the original 8% solution to the amount of alcohol in the final 80% solution:
0.08x(10 +x)
Here, x represents the amount of pure alcohol that we need to add to the 10 cm³ of 8% solution to obtain the desired 80% solution.
The left-hand side of the equation represents the amount of alcohol in the original solution (which is 8% alcohol), while the right-hand side represents the amount of alcohol in the final solution (which is 80% alcohol).
Now we can solve for x:
0.08 x (10 + x) = 0.08x(10+x)
0.2x = 7.2
x = 36 cm³.
Therefore, the pharmacist must add of pure alcohol to the of 8% alcohol solution to obtain an 80% alcohol solution.
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Elena is trying to figure out how many movies she can download to her hard
drive. The hard drive is supposed to hold 500 gigabytes of data, but 58
gigabytes are already taken up by other files. Each movie is 8 gigabytes. Elena
wrote the inequality 8x + 58 ≥ 500 and solved it to find the solution x ≥ 55. 25.
4a) Explain how you know Elena made a mistake based on her solution.
4b) Fix Elena's inequality and explain what each part of the inequality represent.
Based on Elena's solution, x represents the number of movies that can be downloaded. However, her inequality is incorrect as it states that the total amount of downloaded data (8x + 58) is greater than or equal to the total capacity of the hard drive (500 gigabytes).
This means that Elena is considering the amount of data taken up by other files in addition to the movies she wants to download. Therefore, her solution of x ≥ 55.25 is incorrect as it would allow Elena to download more movies than the remaining capacity of the hard drive.
The corrected inequality should be 8x ≤ 442, as the remaining capacity of the hard drive is 500 - 58 = 442 gigabytes. This means that Elena can download a maximum of 55 movies (8 x 55 = 440), leaving 2 gigabytes of space remaining on the hard drive. Therefore, each part of the inequality represents the total amount of data used by the movies downloaded (8x) and the remaining capacity of the hard drive (442).
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In order for a triangle to be acute, what relationship must c2 have with a2 + b2?
group of answer choices
c2>a2+b2
c2
c2=a2+b2
In order for a triangle to be acute, the relationship that c² must have with a² + b² is c² < a² + b².
An acute triangle is a triangle in which all three angles are acute angles, which means they are less than 90 degrees. In other words, an acute triangle is a triangle with three acute angles.
To understand why the relationship between c^2 (the square of the longest side) and a^2 + b^2 (the sum of the squares of the other two sides) is important in determining whether a triangle is acute, we need to delve into the concept of the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, it can be expressed as c^2 = a^2 + b^2, where c represents the hypotenuse, and a and b represent the other two sides.
In an acute triangle, the sum of the squares of the two shorter sides must be greater than the square of the longest side.
This can be visualized as follows: If we were to draw a right triangle with the shorter sides represented by segments a and b, and the longest side represented by segment c, the acute triangle would be formed by making the length of segment c shorter than the length determined by the Pythagorean theorem. This ensures that the angle opposite to the longest side remains acute.
On the other hand, if c^2 were equal to a^2 + b^2, we would have a right triangle, not an acute triangle. If c^2 were greater than a^2 + b^2, we would have an obtuse triangle since the angle opposite to the longest side would be greater than 90 degrees.
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What is the multiplicity of the zero of the polynomial function that represents the volume of a sphere with radius x+5
The graph of the function will touch the x-axis at x = -5, but not cross it, and the behavior of the graph near x = -5 will be determined by the degree of the zero (which is 3 in this case).
The polynomial function that represents the volume of a sphere with radius x+5 is given by:
[tex]V(x) = (4/3)\pi (x+5)^3[/tex]
To find the multiplicity of the zero, we need to factor out the (x+5) term from the polynomial:
V(x) = (4/3)π(x+5)(x+5)(x+5)
We can see that the zero is x = -5, and it has a multiplicity of 3, since there are three factors of (x+5) in the polynomial.
This means that the graph of the function will touch the x-axis at x = -5, but not cross it, and the behavior of the graph near x = -5 will be determined by the degree of the zero (which is 3 in this case).
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6 Evaluate without using calculators. -4(-2)+(-12)÷(+3)+-20+(+4)+(-6
Answer:
Its 12
Step-by-step explanation:
1. Following PEMDAS, we first solve the equation inside the parentheses.
-4(-2) = 8
-12 ÷ 3 = -4
-20 + 4 = -16
2. Now, we have the following expression:
8 + (-4) + (-16)
3. Again, following PEMDAS, we solve the equation inside the parentheses first.
8 + (-4) + (-16) = 8 - 4 - 16
4. Finally, we solve the equation from left to right.
8 - 4 - 16 = 8 - (4 + 16)
8 - (20) = -12
Therefore, the value of the expression is -12.
7 women and 7 men are on the faculty in the mathematics department at a school. how many ways are there to select a committee of five members of the department if at least one woman must be on the committee?
In 1981 number of ways a committee of five members of the department if at least one woman must be on the committee.
By choosing some items from a set and creating subsets, permutation and combination are two approaches to express a collection of things. It outlines the numerous configurations for a certain set of data. Permutations are the selection of data or objects from a set, whereas combinations are the order in which they are represented. Both ideas are critical to mathematics.
The number of ways of picking 5 from 14 is what the question is actualy asking minus combinations of 5 from 7 , because there must be 1 woman
so 5 from 14 is given by :
= [tex]\frac{14*13*12*11*10}{5*4*3*2*1}[/tex]
which is :
14 x 13 x 11 = 2002
combinations of 5 from 7 is :
(9x8x7x6x5) / (5x4x3x2x1)
[tex]\frac{7*6*5*4*3}{5*4*3*2*1}[/tex]
which is :
7 x 3= 21
so the final answer is 2002 - 21 = 1981.
Therefore, in 1981 ways a committee of five members of the department if at least one woman must be on the committee.
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A triangular prism has a net as shown below.
4m
5m
5m
3m
What is the surface area of this triangular prism?
20 points if you help me out!
Answer: 72 m²
Step-by-step explanation:
50 POINTS ASAP Triangle 1 and triangle 2 are similar right triangles formed from a ladder leaning against a building.
Triangle 1 Triangle 2
The distance, along the ground, from the bottom of the ladder to the building is 12 feet. The distance from the bottom of the building to the point where the ladder is touching the building is 18 feet. The distance, along the ground, from the bottom of the ladder to the building is 8 feet. The distance from the bottom of the building to the point where the ladder is touching the building is unknown.
Determine the distance from the bottom of the building to the point where the ladder is touching the building for triangle 2.
27 feet
18 feet
12 feet
5 feet
The distance where the ladder is touching the building for triangle 2 is 12 ft
Determining the distance from the bottom of the building to the pointFrom the question, we have the following parameters that can be used in our computation:
Ladder 1
Distance along the ground = 12 ft
Distance touching the ladder = 8 ft
Ladder 2
Distance along the ground = 18 ft
Distance touching the ladder = x
Using proportion of similar triangles, we have
x : 18 = 8 : 12
Express as fraction
x/18 = 8/12
So, we have
x = 18 * 8/12
Evaluate
x = 12
Hence, the distance where the ladder is touching the building for triangle 2 is 12 ft
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Answer:
12?
Step-by-step explanation:
Not too sure! I am in the middle of taking the test right now though
Triangles UVW and WXY are similar right triangles. Which proportion can be used to show that the slope of uw is equal to the slope of wy ?
a: 1-2 -3 - ( -1 )
9-8 1-2
b: 2-8 -1-2
1-9 -3-1
c: 1-9 -3-1
2-8 -1-2
d: 2 - ( -1 ) -1 - ( -3 )
9 -8 2 -1
The proportion that can be used to show that the slope of uw is equal to the slope of wy is B) 2-8 -1-2.
Let the angles opposite to sides UV, VW, and WU be denoted by theta1, theta2, and theta3, respectively, in triangle UVW. Similarly, let the angles opposite to sides WY, YX, and XW be denoted by theta4, theta5, and theta6, respectively, in triangle WXY. Since triangles UVW and WXY are similar right triangles, we have:
theta1 = theta4 = 90 degrees (both triangles are right triangles)
theta2 = theta5 (corresponding angles in similar triangles are equal)
theta3 = theta6 (corresponding angles in similar triangles are equal)
UW/VW = WY/YX (sides in similar triangles are proportional)
The slope of uw is given by the rise over run or (VU - UW)/(WV), and the slope of wy is given by the rise over run or (XY - WY)/(YW). Using the similar triangles proportion, we can substitute UW/VW = WY/YX and simplify to get (VU - UW)/(WV) = (XY - WY)/(YW). This equation shows that the slope of uw is equal to the slope of wy.
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Rote tells the little monsters to do an overhead press every $12$ seconds and a squat every $30$ seconds. (For example, they should do their first squat $30$ seconds into the drill.)
How many times during the $200$ second drill should the little monsters do an overhead press and a squat at the same instant?
The number of times that the little monsters will do an overhead press and a squat at the same instant during the drill is: 3 times
How to solve Prime Factorization Problems?We are told that, Rote tells the little monsters to carry out an overhead press every 12 seconds and then also a squat every 30 seconds.
Thus, prime factorization of 12: 2² x 3.
Thus, prime factorization of 30: 2 x 3 x 5.
For us to get the least common multiple, we will have to find the highest power of each of the prime factors that show up in either factorization. Therefore, we can say that the smallest common multiple of 12 and 30 are: 2² x 3 x 5 = 60
This tells us that the little monsters will carry out an overhead press and then a squat at the same instant for every 60 seconds.
For us to find how many times this will occur during the 200-second drill, we will then divide 200 by 60:
200 ÷ 60 = 3 remainder 20
This tells us that there will exist 3 complete cycles of both of the exercises during the 200-second drill, with an additional 20 seconds left over.
Therefore, we can say that the little monsters will do an overhead press and a squat at the same instant 3 times during the drill.
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Bilquis is trying to find the height of a radio antenna on the roof of a local building. She stands at a horizontal distance of 29 meters from the building. The angle of elevation from her eyes to the roof (point A) is 17°, and the angle of elevation from her eyes to the top of the antenna (point B) is 31º. If her eyes are 1. 51 meters from the ground, find the heigh[of the antenna (the distance from point A to point B). Round your answer to the nearest meter if necessary.
The height of the antenna is approximately 17 meters.
To solve for the height of the antenna, we need to use trigonometry. Let's call the height of the antenna "h".
First, we need to find the distance from point A to point B. We can use the angle of elevation from her eyes to the roof (17°) and the horizontal distance from the building (29m) to find the height of the building.
tan(17°) = height of building / 29m
height of building = 29m * tan(17°)
height of building = 8.20m
Now we can use the height of the building and the angle of elevation from her eyes to the top of the antenna (31°) to find the distance from point A to point B.
tan(31°) = h / 29m + 8.20m
h = (29m + 8.20m) * tan(31°)
h = 15.51m
Finally, we need to add the height of Bilquis' eyes to the height of the antenna to find the total height.
total height = h + 1.51m
total height = 17.02m
Therefore, the height of the antenna is approximately 17 meters.
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PLS HELP! and actually answer the question please
Step-by-step explanation:
First start with the graph of y = | x|
then shift it RIGHT one unit
| x -1 |
then shift it DOWN one unit
y= |x-1| -1
Evaluate the following limit. Use l'Hôpital's Rule when it is convenient and applicable. sin lim 5 X400 :) X lim 5 X00 64--m-0 - sin)=(Type an exact answer)
Let's evaluate the limit using l'Hôpital's Rule when it is convenient and applicable:
The evaluated limit using l'Hôpital's Rule is 5.
Given limit,
lim (x -> 0) (sin(5x) / x)
Since both the numerator and denominator approach 0 as x approaches 0,
we can apply l'Hôpital's Rule.
Step 1: Differentiate the numerator and the denominator with respect to x.
- Derivative of sin(5x) with respect to x: 5*cos(5x)
- Derivative of x with respect to x: 1
Step 2: Apply l'Hôpital's Rule:
lim (x -> 0) (5*cos(5x) / 1)
Step 3: Evaluate the limit:
As x approaches 0, cos(5x) approaches cos(0) = 1.
Therefore, the limit is: 5*1 = 5
So, the evaluated limit using l'Hôpital's Rule is 5
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Aura builds model airplanes. Her first model airplane is 4
feet long
She wants her next model airplane to be
4
as long as the first
How long will her next modhi airplane be?
1
ft
12
B
1
4
12
ft
7
4
12
ft
D
17
ft
The next model airplane will be 16 feet long.
How long will Aura's next model airplane be if she wants it to be four times as long as her first model airplane which is 4 feet long?To find the length of Aura's next model airplane, we need to multiply the length of her first model airplane by 4, since she wants the next one to be 4 times as long as the first. Therefore, the length of her next model airplane would be:
4 feet (length of first model airplane) x 4 = 16 feet
So, the length of her next model airplane would be 16 feet.
Note that this assumes that Aura's first model airplane is the baseline for measurement and that the "4" in the question refers to a factor of 4, not an additional 4 feet. If the question were interpreted differently, the answer may be different.
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Solve by graphing:
(x - 2)² = 9
Thanks!
Can someone help me asap? It’s due today
Step-by-step explanation:
the answer will be "15" according to the question.
Aiden gave each member of his family a playlist of random songs to listen to and asked them to rate each song between 0 and 10. He compared his family’s ratings with the release year of each song and created the following scatterplot:
What would the linear equation be?
The linear equation from the given scatterplot will be y = -0.1x + 9.
On the given scatterplot we have the song released details on the x-axis and the average rating of the songs by the family members on the y-axis.
To get the linear equation from the given scatterplot we have to find the y-intercept of the equation.
The general form of the equation is y = mx + c
here, m is the slope and c is the y-intercept.
By, the given graph we can say that y is intercepting at the value '9'. So, the y-intercept is 9.
To find the slope we have to take two points,
Let's take two points as (1970, 7) and (1990, 5).
From the points slope = (5-7)/(1990-1970)
= -2/20 = -1/10 = -0.1
So, the equation from the given scatterplot is y = mx+c
So, y = -0.1x + 9.
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