The radius of the circular fountain is approximately 17 feet.
The formula for the volume of a circular fountain is given by V = πr^2h, where V is the volume, r is the radius, and h is the depth. In this case, we are given that the depth of the fountain is 3 feet and the volume is 1800 cubic feet. So we can plug in these values into the formula and solve for r as follows:
1800 = πr^2(3)
Simplifying this equation, we get:
r^2 = 600/π
Taking the square root of both sides, we get:
r = sqrt(600/π)
Using a calculator to approximate the value of sqrt(600/π), we get:
r ≈ 17
Therefore, the radius of the circular fountain is approximately 17 feet when rounded to the nearest whole number.
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Please help I need this done ASAP
Answer:
Domain is all x values
Range is all y values
Step-by-step explanation:
Your image is not clear enough for me to see the x or y coordinates so hope that helps you to figure it out on your own
HELPPPPPPP PLEASEEEE
Answer:
The first box and whisker plot
Step-by-step explanation:
A box and whisker plot gives you the five number summary for a set of data. The five number summary is
The minimum/lowest value (looks like the top of capital T turned sideways and is the leftmost part of the box-and-whisker plot The first quartile or Q1, representing 25% of the data (the first point represented in the "box" of the plot and serves as an endpoint of the box)The median or Q2, representing 50%/the middle of the data (the line that splits the box into two parts/the line in the middle of the box)The third quartile or Q3, representing 75% of the data (the last point represented in the "box" of the plot and serves as another endpoint of the box)The maximum/highest value (also looks like the top of capital T turned sideways and is the rightmost part of the box-and-whisker plotMaximum and minimum:
We know from the data that the minimum value is 100 and the maximum value is 200. However, because both boxes available as answer choices have the correct minimum and maximum, we'll need to find more data.
Median:
We can start finding the median first by arranging the data from the least to greatest. Then, we find the middle of the data. Because there are 9 points and 9 is odd, we know that there will be 4 points to the left of the median and 4 points to the right of the median:
100, 100, 120, 120, 150, 165, 180, 180, 200
150 has 4 numbers both on its left and right sides so its the median.
Because both of the plots available as answer choices have the correct median, we we'll need to find more data.
First Quartile/Q1:
In order to find Q1, we must find the middle number of the four numbers to the left of the median.
Because we have an even number of points, we will get two middle numbers, 100 and 120. To find the middle of all four points, we average these two numbers:
(100 + 120) / 2 = 220 / 2 = 110
Only the first box has the accurate Q1 value, so it's our answer.
We don't have to find Q3, since both boxes have the correct Q3, but only the first box has the correct minimum, correct Q1, correct median, correct Q3, correct maximum.
(1 point) Calculate T..T,, and n(u, v) for the parametrized surface at the given point. Then find the equation of the tangent plane to the surface at that point. (u, v) = (24 + 0,u - 40, 8u): u = 3. V = 9 T, T, = n(u, v) = . The tangent plane: = 92
Given points:
(u, v) = (24 + 0,u - 40, 8u): u = 3. V = 9 T, T, = n(u, v)
the equation of the tangent plane at the point (u, v) = (24, 9) is:-8x + z = -183T
Process of finding equation:
To start, let's find T..T,, which represents the magnitude of the tangent vector at the given point:
T..T, = ||n(u, v)|| = ||n(24, -31, 72)|| = ||<48, -62, 144>|| = sqrt(48^2 + (-62)^2 + 144^2) = sqrt(11668) ≈ 108.03
Next, let's find the normal vector n(u, v) at the given point:
n(u, v) =
where f_u and f_v are the partial derivatives of the surface equation with respect to u and v, respectively.
In this case, we have:
f(u, v) = (24 + 0,u - 40, 8u)
f_u = <0, 1, 8>
f_v = <1, 0, 0>
Therefore, at the point (u, v) = (24, 9), we have:
n(u, v) = <0, 1, 8> x <1, 0, 0> = <-8, 0, 1>
Finally, let's find the equation of the tangent plane at the point (u, v) = (24, 9). The equation of a plane can be written as:
Ax + By + Cz = D
where A, B, and C are the components of the normal vector, and D can be found by plugging in the coordinates of the point on the plane. In this case, we have:
A = -8
B = 0
C = 1
D = -8(24) + 1(9) = -183
Therefore, the equation of the tangent plane at the point (u, v) = (24, 9) is:
-8x + z = -183
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An oil tanker and a cruise ship leave port at the same time and travel straight-line at 32 mph and 46 mph, respectively. Two hours later, they are 63 miles apart. What is the angle between their courses?
The angle between their courses is 42.02°.
How to calculate angle between 2 moving bodiesIt is important to first find the distance between them after the 2 hours of travel.
Recall the formula:
speed = distance/time
Make distance the subject of the formula
distance = speed x time
For the oil tanker,
given the following:
speed = 32mph
time = 2hr
distance = 32 mph x 2 hours = 64 miles
For the cruise ship,
given the following:
speed = 46 mph,
time = 2 hr
distance = 46 mph x 2 hours = 92 miles
So after two hours of travel, the two vessels are 63 miles apart. This means that they are forming a triangle with the distance between them as the longest.
Now we need to find the angle between the two vessels' courses by using the Cosine rule:
Recall that
a² = b² + c² -2bc Cos A
Let C be the angle between the oil tanker and cruise ship
then we can rewrite the equation as:
c² = a² + b² -2bc Cos C
where
a = 64miles (distance of oil tanker)
b = 92miles (dsitance of cruise ship)
c = 63miles (distance between the vessels)
C = angle between the vessels
Plug in the values to the equation
63² = 64² + 92² - 2(64)(92) Cos C
3969 = 4096 + 8464 - 11776 Cos C
3969 = 12560 - 11776 Cos C
Collect like terms
3969 - 12560 = - 11776 Cos C
8591 = 11776 Cos C
Cos C = 8591/11776
Cos C = 0.7295
Apply the inverse Cosine formula
C = Cos⁻¹ (0.7295)
C = 42.02°
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What are the 2 terms that are associated with input?
The two terms that are commonly associated with input are input device and input data.
Input Bias are essential factors of computer systems as they enable druggies to interact with the machine and give data or information that the computer processes to produce affair. There are several types of input bias, each designed to feed to specific requirements and conditions. For illustration, a keyboard is a common input device used to input textbook and commands, while a microphone is used to input audio data.
Input data, on the other hand, can come in colorful forms and formats. It can be entered manually by a stoner or automatically collected by detectors, bias, or other systems. Input data can also be stored in colorful train formats, similar as textbook, audio, videotape, images, or databases. It's reused by the computer system using algorithms, software programs, and tackle factors to induce affair data, results, or conduct.
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Evaluate the repeated integral: lolla (-xy + 2 z) dz dy dx a) O 15 b) 60 c) 30 d) 36 e) O72 f) O None of these.
The evaluation of the repeated integral is None of these. (option f)
The repeated integral given is ∫∫∫(-xy + 2z) dz dy dx over the region lolla. This means that you need to integrate the function (-xy + 2z) with respect to z, then with respect to y, and finally with respect to x over the region lolla.
To evaluate this integral, you can use the method of iterated integrals. First, integrate (-xy + 2z) with respect to z, treating x and y as constants:
∫∫(-xy + 2z) dz = -xyz + z² + C
where C is the constant of integration.
Next, integrate the result of the first integral with respect to y, treating x as a constant:
∫[-xyz + z² + C] dy = -xyz + y[-xyz + z² + C] + D
where D is the constant of integration.
Finally, integrate the result of the second integral with respect to x:
∫[-xyz + y(-xyz + z² + C) + D] dx = (-1/2) x² yz + xy(-xyz + z² + C) + Dx + E
where E is the constant of integration.
Now, you need to evaluate this expression over the region lolla. Without further information about the limits of integration for each variable, it is not possible to determine the exact value of this integral.
Therefore, the correct answer is f) None of these.
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Under her cell phone plan Yaritza pays a flat cost of $41 and 50 Cent per month and five dollars per gigabyte she wants to keep her bill under $60 per month which inequality can be used to determine ask the minimum number of gigabytes Yahritza can use while staying within her budget
Answer:3 gigabytes of storage.
Step-by-step explanation: Because you start at $41.50 and add 5 is $46.60 and then add 5 again and you get $51.50 then add 5 more you get $56.50.
The function f(z) = 1 + 6x + 96x^-1 has one local minimum and one local maximum.
This function has a local maximum at x?
The function f(z) = 1 + 6x + 96x^-1 has one local minimum and one local maximum. This function has a local maximum at x. The function f(x) = 1 + 6x + 96x^(-1) has a local maximum at x = -4.
To find the local maximum of the function f(x) = 1 + 6x + 96x^(-1), we first need to find the critical points. We do this by finding the first derivative of the function and setting it equal to zero.
Step 1: Find the derivative of the function
f'(x) = d/dx (1 + 6x + 96x^(-1))
f'(x) = 6 - 96x^(-2)
Step 2: Set the derivative equal to zero and solve for x
6 - 96x^(-2) = 0
Step 3: Solve for x
96x^(-2) = 6
x^(-2) = 6/96
x^(-2) = 1/16
x^2 = 16
x = ±4
Step 4: Determine which of the critical points is a local maximum. To do this, we will examine the second derivative of the function.
f''(x) = d^2/dx^2 (1 + 6x + 96x^(-1))
f''(x) = 192x^(-3)
Now we will evaluate the second derivative at each critical point:
f''(4) = 192(4)^(-3) = 3 > 0, so x = 4 is a local minimum.
f''(-4) = 192(-4)^(-3) = -3 < 0, so x = -4 is a local maximum.
Therefore, the function f(x) = 1 + 6x + 96x^(-1) has a local maximum at x = -4.
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PLEASE HELP EM I WIL GIVE BRAINLIEST TO THE FIRST CORRECT ANSWER EHLP ME FAST PLEASE
[tex]a = \sqrt{ {8}^{2} - {6}^{2} } \\ \\ = \sqrt{64 - 36 }\\ \\ =\sqrt{ 28} \\ \\ = \sqrt{4 \times 7} \\ \\ = 2 \sqrt{7} [/tex]
During their team meeting, both managers shared their findings. Complete the statement describing their combined results.
Select the correct answer from each drop-down menu.
The initial number of video views was ____ the initial number of site visits, and the number of video views grew by _____ the number of site visits.
The difference between the total number of site visits and the video views after 5 weeks is _____.
More than
the same as
fewer than
a smaller factor than
the same factor as
a larger factor than
20,825
52,075
15,625
36,450
The initial number of video views was fewer than the initial number of site visits, and the number of video views grew by a larger factor than the number of site visits. The difference between the total number of site visits and the video views after 5 weeks is 20,825.
During their team meeting, both managers shared their findings. Complete the statement describing their combined results.
The initial number of video views was fewer than the initial number of site visits, and the number of video views grew by a larger factor than the number of site visits.
The difference between the total number of site visits and the video views after 5 weeks is 20,825.
Therefore the correct answer are fewer, larger factor than and 20,825.
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In order to estimate the mean 30-year fixed mortgage rate for a home loan in the United States, a random sample of 10 recent loans is taken. The average calculated from this sample is 6. 40%. It can be assumed that 30-year fixed mortgage rates are normally distributed with a population standard deviation of 0. 5%. Compute 95% and 99% confidence intervals for the population mean 30-year fixed mortgage rate
The 95% confidence interval for the population mean 30-year fixed mortgage rate is (6.091%, 6.709%), and the 99% confidence interval is (5.993%, 6.807%).
To estimate the mean 30-year fixed mortgage rate for a home loan in the United States using a random sample of 10 recent loans with an average of 6.40% and a population standard deviation of 0.5%, you can compute the 95% and 99% confidence intervals as follows:
1: Identify the sample mean (x), sample size (n), and population standard deviation (σ).
x = 6.40%
n = 10
σ = 0.5%
2: Calculate the standard error (SE) using the formula SE = σ/√n.
SE = 0.5%/√10 ≈ 0.158%
3: Determine the critical z-values for 95% and 99% confidence intervals.
For a 95% confidence interval, z = 1.96 (from the z-table)
For a 99% confidence interval, z = 2.576 (from the z-table)
4: Calculate the margin of error (ME) using the formula ME = z * SE.
For 95% CI,
ME = 1.96 * 0.158% ≈ 0.309%
For 99% CI,
ME = 2.576 * 0.158% ≈ 0.407%
5: Compute the confidence intervals by adding and subtracting the ME from the sample mean.
95% CI:
(6.40% - 0.309%, 6.40% + 0.309%) = (6.091%, 6.709%)
99% CI:
(6.40% - 0.407%, 6.40% + 0.407%) = (5.993%, 6.807%)
So, the 95% confidence interval and 99% confidence interval is (6.091%, 6.709%) and (5.993%, 6.807%) respectively.
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Which of the following is a typical characteristic of debit cards? (1 point)
O Your bank may charge you a very large fee each time you use one.
O When you buy something from a store, you may be offered a discount if you open one with the store.
O They usually charge a lower interest rate than credit cards.
They are tied directly to your bank account.
A typical characteristic of debit cards is expressed by option D, they are tied directly to your bank account, since money has to be deducted for a purchase to be made.
How debit cards functionWhen you use a debit card to make a purchase, the money is deducted directly from your bank account. Debit cards do not typically charge interest like credit cards do, so C is not correct.
A is also not correct since most banks do not charge a fee for using a debit card, although some may charge fees for using an out-of-network ATM or overdraft fees if you spend more than you have in your account. B is not a typical characteristic of debit cards either, as there is generally no connection between a store's loyalty program or discounts and using a debit card for payment.
Therefore, it is possible to conclude that debit cards have option D as their characteristic. Since they require money in the bank to make the purchase, they are connected to one's account.
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1. $4,076.92
2. Assets = Liabilities + Equity
3. They can be used to make online purchases.
4. by using an emergency fund for this unplanned expense
5. a fund with a minimum investment that tracks the value of cash
6. 1930s
7. Banks because they are highly regulated by the government, so the loan terms will not be predatory.
8. their vacation cabin
9. You should shop around for the best overall deal.
10. If you use a credit card, it is easy to run up huge debts.
11. They are tied directly to your bank account.
12. preventative care
13. $100,000 per person bodily injury, $300,000 per incident for bodily injury, $50,000 for property damage
14. how long the coverage lasts, how much the premium costs, and the cash value
15. 1-year renewable group term life
16. Identity thieves can intercept unencrypted data being sent to Wi-Fi hot spots.
17. Wells Fargo employees were opening unauthorized deposit and credit accounts for its customers.
18. 2 year in state community college degree
19. tuition assistance
20. -a sundae, -movie tickets
Explanation: All of these answers are correct!
Personal Finance Semester Exam
5/11/2023
A track has the dimensions shown.
36.5 m
ISTAN
SAMANT
men komm
84.4 m
Ta
inside of track
outside of track
. The track has 8 lanes
• Each lane is 2.1 meters wide
36.5 m
O
TI
16. To the nearest tenth of a meter, what is
the perimeter of the outside of the
track?
Byp
*REQUIRED
ANA
1
√x
Sign out
Answer:
Step-by-step explanation:
Shaunda measures the diameter of a ball as 12 in. How many cubic inches of air does this ball hold? Round your answer to the nearest tenth
To find the volume (cubic inches) of the ball, we first need to find its radius, which is half the diameter.
Radius = 12 in / 2 = 6 in
Now we can use the formula for the volume of a sphere:
Volume = (4/3) x π x radius^3 cubic inches
Volume = (4/3) x π x 6^3
Volume ≈ 904.8 cubic inches
So the ball holds approximately 904.8 cubic inches of air. Rounded to the nearest tenth, the answer is 904.8.
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CopyCat Pear-Apple wants to get into the tablet business. They want to make tablets similar to Apple's iPad Mini 2. If Pear-Apple applies a scale factor of 1. 25 to the iPad Mini 2 to make a new tablet called the BigMini, what would be the new volume in cubic inches?
The new volume of the BigMini tablet is about 22.18 cubic inches.
How to calculate the new volume of the BigMini tablet?The volume of a three-dimensional object, such as a tablet, is calculated by multiplying its length, width, and height.
Assuming that the scale factor of 1.25 is applied uniformly to all three dimensions of the iPad Mini 2, the new dimensions of the BigMini tablet would be:
Length: 1.25 x 7.87 inches = 9.84 inches
Width: 1.25 x 5.3 inches = 6.63 inches
Height: 1.25 x 0.29 inches = 0.36 inches
The new volume of the BigMini tablet can be calculated as:
9.84 x 6.63 x 0.36 = 22.18 cubic inches
Therefore, the new volume of the BigMini tablet would be approximately 22.18 cubic inches.
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You have $20 to spend. You go to the store and buy a bouncy ball for an unknown amount of money and then you buy a glider airplane for $3. If you have $15 left over, how much did you spend on the bouncy ball?
Answer: Let's start by subtracting the cost of the glider airplane from the total amount of money you started with:
$20 - $3 = $17
We know that you spent $15 of that $17 on the bouncy ball, since you had $15 left over after buying both items:
$17 - $15 = $2
Therefore, you spent $2 on the bouncy ball.
Answer: $2.
Step-by-step explanation:
Jessica has 300 cm of matenal. She uses 12. 6 cm to make a nght triangular prism She wants to make a second prism that is a
dilation of the first prism with a scale factor of 3
How much more material does Jessica need in order to make the second prism?
Select from the drop-down menu to correctly complete the statement
cm of material to make the second prism
Jessica needs an additional Choose
To make the second prism, Jessica needs an additional 25.2 cm of material.
To answer your question, since Jessica wants to create a second triangular prism with a scale factor of 3, she will need 3 times the material used for the first prism.
She used 12.6 cm for the first prism, so for the second prism, she would need 12.6 cm × 3 = 37.8 cm of material.
Jessica already has 300 cm of material, so to find out how much more she needs, subtract the amount used for the first prism: 37.8 cm - 12.6 cm = 25.2 cm.
Jessica needs an additional 25.2 cm of material to make the second prism.
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Pls answer this, 5 points and brainliest for the one who answers first.
Answer: A
Step-by-step explanation:
It's A because our function of f is multiplied by 3.
Since our y intercept is 1, and we are multiplying the function f by 3,
our new y intercept is 3, meaning it is A.
Another way to check this is by using the other two points on your graph.
Please give brainliest + have a good afternoon.
Answer:
Step-by-step explanation:
Your original function has points at
0, 1
1, 2
2,4
if you stretched it by 3, multiply your y by 3
new function:
0, 3
1, 6
2, 12
A tank in the shape of a hemisphere has a diameter of 8 feet. If the liquid that fills the tank has a density of 86 pounds per cubic foot, what is the total weight of the liquid in the tank, to the nearest full pound?
Answer:
209.07 pounds
Step-by-step explanation:
radius= 8÷2=4 feet
volume of hemisphere=((4/3)×(22/7)×r^3)/2
=134.09 cubic feet
Mass=density × volume
=86×134.09
=209.07 pounds
A line that passes through the point (x,y) with a y-intercept of b and a slope of m can be represented by the equation y = mx + b.
Joe drew a line on the coordinate plane that passes through the point (-10,52) and has a slope of -6.5. The y-intercept of the line is
The y-intercept of the line is -13.
To find the y-intercept of the line, we can use the slope-intercept form of the equation of a line: y = mx + b,
where m is the slope and b is the y-intercept.
Given that the line passes through the point (-10, 52) and has a slope of -6.5, we can substitute these values into the equation:
52 = -6.5(-10) + b
Simplifying the equation:
52 = 65 + b
To isolate b, we subtract 65 from both sides:
52 - 65 = b
Simplifying further:
b = -13
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I need help on this question, and please explain how you did it.
The expression for AB in terms of x and √3 is:
AB = x√3.
What is an expression?An expression in mathematics is a combination of numbers, variables, and/or operators that represents a mathematical relationship or quantity. It may contain constants, variables, coefficients, and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation. Expressions are often used to describe or represent real-world situations, and can be simplified, evaluated, or manipulated using algebraic rules and properties.
In the given question,
In a right triangle ABC, if sin B = 0.5, then we know that:
sin B = opposite / hypotenuse
So, we can write:
0.5 = AB / CB
We also know that:
CB² = AB² + AC²
Substituting the value of AC, we get:
CB² = AB² + (3x)²
CB² = AB² + 9x²
Now, we can substitute the value of CB² from the first equation:
(AB / 0.5)² = AB² + 9x²
4AB² = AB² + 9x²
3AB² = 9x²
AB² = 3x²
The expression for AB in terms of x and √3 is:
AB = x√3
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The table shows the amount of money raised during a car wash for charity.
Number of Cars Washed Money Raised
3 $43.50
13 $279.50
18 $405.00
Which statement is true?
A. The group raised $14.50 per car.
B. The group raised $21.50 per car.
C. The group raised $22.50 per car.
D. The relationship is not a direct proportion.
Answer:
The correct answer is D. The relationship is not a direct proportion.
We can see that the money raised is not directly proportional to the number of cars washed. For example, when the number of cars washed is doubled from 3 to 13, the money raised is not doubled from $43.50 to $87.00. Instead, it is increased by a factor of 6.5, from $43.50 to $279.50. Similarly, when the number of cars washed is increased by 5 from 13 to 18, the money raised is increased by a factor of 1.4, from $279.50 to $405.00.
This suggests that the amount of money raised is not simply a linear function of the number of cars washed. Instead, it is likely a more complex function that takes into account other factors, such as the time of day, the weather, and the location of the car wash.:
Barbara’s Bigtime Bakery baked the world’s largest chocolate cake. (It was also the world’s worstcake, as 343 people got sick after eating it. ) The length was 600 cm, the width 400 cm, and the height 180 cm. Barbara and her two assistants, Boris and Bernie, applied green peppermint frosting on the four sides and the top. How many liters offrosting did they need for this dieter’s nightmare? One liter of green frosting covers about 1200 cm²
The total liters of frosting needed is 500, under the condition that the length was 600 cm, the width 400 cm, and the height 180 cm.
In order to evaluate the amount of frosting needed, we have to evaluate the surface area of the cake. The surface area of the cake is the summation of the areas of all its sides.
Here the area of each side is equivalent to its length multiplied by its width. Then the area of the given top is equivalent to its length multiplied by its width.
Then the evaluated surface area of the cake is
2 × (length × height + width × height) + length × width
= 2 × (600 cm × 180 cm + 400 cm × 180 cm) + 600 cm × 400 cm
= 2 × (108000 cm² + 72000 cm²) + 240000 cm²
= 2 × 180000 cm² + 240000 cm²
= 600000 cm²
Hence, one liter of green frosting covers about 1200 cm².
600000 cm² / 1200 cm² per liter = 500 liters
Therefore, Barbara and her assistants needed 500 liters of frosting for their dieter's nightmare.
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Write an appropriate and interesting word for 15000 and 3000.solve it
An appropriate and interesting word for 15,000 is "fifteen thousand," while for 3,000, it is "three thousand."
To solve this question, we need to write the numbers 15,000 and 3,000 as words. To do this, first, we look at the place values of each digit. In 15,000, the "15" is in the thousands place, so we write it as "fifteen thousand." Similarly, in 3,000, the "3" is in the thousands place, so we write it as "three thousand."
By doing this, we have expressed the numbers using their word forms, which helps in understanding and communicating numerical values more effectively, especially in written or spoken contexts. Remember, when writing large numbers as words, we use the place values to guide us in expressing them accurately and understandably.
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Question 6 of 20 :
Select the best answer for ige question. 6. Simplify (4x 4)-3. O B. 2
O C. -8x12
0D. -64x9
The correct answer is (C) -8x12.
To simplify (4x^4)^-3, we use the power of a power rule which states that (a^m)^n = a^(mn), where a is a non-negative number and m and n are integers. Applying this rule, we get:
(4x^4)^-3 = 4^(-3) x^(4 x -3) = (1/64)x^(-12) = -8x^12 (using the negative exponent rule, which states that a^(-n) = 1/a^n)
Therefore, the simplified form of (4x^4)^-3 is -8x^12.
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What is the measure of “?”
The part of the figure marked ? is solved to be
113 degrees
How to find the question markThe part of the circle marked by a question mark is solved using the relation as shown below
given angle formed by the tangents = 180 degrees - minor arc GE
information given in the problem includes
given angle formed by the tangents = 67 degrees
minor arc GE = ?
plugging in these values results to
67 degrees = 180 degrees - ?
rearranging the equation
? = 180 degrees - 67
? = 113
hence the required side is 113 degrees
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What is the sine ratio for angle A?
A. 10/8
B. 8/10
C. 6/10
D. 6/8
Answer:
C. 6/10
Step-by-step explanation:
Sin A = 6/10
Find two acute angles that satisfy the equation sin(3x + 9) = cos(x + 5). check that your answers make sense.
The equation sin(3x + 9) = cos(x + 5) has no solutions in the set of acute angles.
What are the acute angles that satisfy sin(3x + 9) = cos(x + 5)?To find two acute angles that satisfy the equation sin(3x + 9) = cos(x + 5), we can use the trigonometric identity cos(x) = sin(π/2 - x) to rewrite the right-hand side of the equation as follows:
sin(3x + 9) = cos(x + 5)sin(3x + 9) = sin(π/2 - x - 5)3x + 9 = π/2 - x - 5 + 2πn or 3x + 9 = x + 5 + 2πn + π (where n is an integer)4x = -4 - 2πn or 2x = -2πn - 4 or 2x = π - 2πn - 4Dividing both sides of the equation by 4, we get:
x = -(1/2)πn - 1
So the solutions are given by:
x = -(1/2)π - 1 and x = -(3/2)π - 1
To check that these solutions make sense, we need to ensure that they are acute angles, i.e., angles that measure less than 90 degrees.
The first solution, x = -(1/2)π - 1, can be written in degrees as:
x ≈ -106.26 degrees
This angle is not acute, so it is not a valid solution.
The second solution, x = -(3/2)π - 1, can be written in degrees as:
x ≈ -286.87 degrees
This angle is also not acute, so it is not a valid solution.
Therefore, there are no acute angles that satisfy the equation sin(3x + 9) = cos(x + 5).
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5. The formula below relates the velocity,
v, of a moving object (in meters per
second), to the kinetic energy, E, of the
object (in joules), and the object's mass,
m (in kilograms).
V=
2.E
m
What is the velocity, in meters
per second, of a bowling ball that
has a mass of 5.5 kilograms and is
producing 2223 joules of kinetic
energy?
The velocity of the bowling ball is approximately 20.104 meters per second.
What is velocity?The pace at which an object's position changes in relation to a frame of reference and time is what is meant by velocity.
The formula given is used to calculate the velocity of a moving object in meters per second, given the object's mass in kilograms and its kinetic energy in joules. The formula is:
V = √(2E/m)
where V is the velocity, E is the kinetic energy, and m is the mass of the object.
To use this formula to find the velocity of the bowling ball, we need to substitute the given values into the formula. The mass of the bowling ball is 5.5 kilograms, and the kinetic energy is 2223 joules. Substituting these values, we get:
V = √(2 × 2223 J / 5.5 kg)
Now, we can simplify the equation:
V = √(404.1818)
Using a calculator, we can find the square root of 404.1818:
V = 20.104 m/s (rounded to three decimal places)
Therefore, the velocity of the bowling ball is approximately 20.104 meters per second.
This formula is useful for calculating the velocity of a moving object when the mass and kinetic energy of the object are known. It can be used in a variety of situations, such as in physics experiments, engineering design, or in understanding the motion of objects in sports.
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5. Rita has a circular hot tub. The hot tub has a circumference 25. 12 feet. It is 3. 5 feet deep.
a. Find the radius of the hot tub. Use 3. 14 for pi
b. How much water can the hot tub hold?
c. The hot tub manual recommends filling the hot tub to 80% of its full capacity. How
much water should rita put in the hot tub in order to follow the recommendation?
a.The radius of the hot tub is 4 feet. b. The amount of water that the hot tub can hold is 176.96 cubic feet c. The amount of water that Rita should put in the hot tub in order to follow the recommendation is 141.57 cubic feet.
a. Find the radius of the hot tub.
Given: Circumference = 25.12 feet
Formula: Circumference = 2 * pi * radius
1: Plug in the given circumference and the value of pi.
25.12 = 2 * 3.14 * radius
2: Solve for the radius.
radius = 25.12 / (2 * 3.14)
radius ≈ 4 feet
So, the hot tub's radius is 4 feet.
b. How much water can the hot tub hold?
Given: Radius = 4 feet, Depth = 3.5 feet
Formula: Volume = pi * radius^2 * depth
1: Plug in the radius, depth, and the value of pi.
Volume = 3.14 * (4^2) * 3.5
2: Calculate the volume.
Volume ≈ 176.96 cubic feet
So, the hot tub can approximately hold 176.96 cubic feet.
c. How much water should Rita put in the hot tub to follow the recommendation?
Given: Recommended capacity = 80% of full capacity
Formula: Recommended water = 0.8 * full capacity
1: Plug in the full capacity (volume) calculated in part b.
Recommended water = 0.8 * 176.96
2: Calculate the recommended water amount.
Recommended water ≈ 141.57 cubic feet
So, Rita should put approximately 141.57 cubic feet of water in the hot tub to follow the recommendation.
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