Final answer:
The correct answer is E) 36sqrt(3), which is the volume of the described solid.
Explanation:
The student's question relates to finding the volume of a solid with unusual properties: it has a circular base and each plane cross-section perpendicular to the x-axis is an equilateral triangle. We will solve this problem using calculus, specifically by integrating the cross-sectional area along the x-axis.
The circular base of the solid defines the limits of integration, as it gives the diameter of the solid which, in turn, represents its height because the triangle is equilateral.
The radius given is 3, so the diameter (and hence the height of the solid) is 6.
The area of an equilateral triangle with side length s is (sqrt(3)/4) * s².
Since the side length of the triangles is equal to the diameter of the base, s is 6.
Therefore, the area of each cross-section is (sqrt(3)/4) * 6².
Now, to find the volume, we integrate the area of each cross-section along the height of the solid. Doing this gives us:
Volume = [tex]int_0^6 ((sqrt(3)/4) * 6^2) dx[/tex]Thus, the correct answer is E) 36sqrt(3), which is the volume of the described solid.
how do you solve log base 3 (27) by hand
The log base 3 of 27 is equal to 3.
To solve log base 3 of 27 by hand, we need to determine the exponent to which the base 3 must be raised to obtain 27.
In this case, 3 raised to the power of what exponent equals 27?
By observing that [tex]3^3[/tex] equals 27, we can conclude that:
log base 3 of 27 = 3.
Therefore, the solution to log base 3 of 27 is 3.
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A triangle with one obtuse angle must also have two acute angles.a. trueb. false
Verify each identity:
(tanx+tany)/(1-tanxtany)=(sinxcosy+cosxsiny)/(cosxcosy-sinxsiny),
Final answer:
The trigonometric identity (tan(x) + tan(y)) / (1 - tan(x)tan(y)) = (sin(x)cos(y) + cos(x)sin(y)) / (cos(x)cos(y) - sin(x)sin(y)) can be verified by using the angle addition formulas for sine and cosine, and by applying the tangent sum formula. Both sides reduce to tan(x + y), confirming the identity.
Explanation:
The given trigonometric identity to verify is:
(tan(x) + tan(y)) / (1 - tan(x)tan(y)) = (sin(x)cos(y) + cos(x)sin(y)) / (cos(x)cos(y) - sin(x)sin(y)).
Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables where both sides of the equation are defined.
Let's verify the given identity step by step. First, rewrite the right-hand side using the angle addition formulas for sine and cosine:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)cos(x + y) = cos(x)cos(y) - sin(x)sin(y)So, the right-hand side can be written as sin(x + y) / cos(x + y). This is equivalent to tan(x + y).
Now, let's examine the left-hand side and apply the tangent sum formula:
tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x)tan(y))As we can see, the left-hand side is exactly the tangent sum formula; therefore it also simplifies to tan(x + y).
Since tan(x + y) is a common form for both sides of the identity, we can conclude that the identity is verified:
(tan(x) + tan(y)) / (1 - tan(x)tan(y)) = sin(x + y) / cos(x + y)
I need help on the bottom problem
Rewrite the following quadratic function in vertex form. Then, determine the axis of symmetry. y=5x^2+15x-2
Answer: The vertex form of the given function is [tex]y=5\left(x+\dfrac{3}{2}\right)^2-\dfrac{53}{4}[/tex] and its axis of symmetry is [tex]x=-\dfrac{3}{2}.[/tex]
Step-by-step explanation: The given quadratic function is
[tex]y=5x^2+15x-2~~~~~~~~~~~~~~(i)[/tex]
We are to rewrite the above function in vertex form and to determine its axis of symmetry.
We have from equation (i),
[tex]y=5x^2+15x-2\\\\\Rightarrow y=5(x^2+3x)-2\\\\\Rightarrow y=5\left(x^2+2\times x\times \dfrac{3}{2}+\dfrac{9}{4}\right)-\dfrac{45}{4}-2\\\\\Rightarrow y=5\left(x+\dfrac{3}{2}\right)^2-\dfrac{45+8}{4}\\\\\Rightarrow y=5\left(x+\dfrac{3}{2}\right)^2-\dfrac{53}{4}.[/tex]
So, the given function is a parabola with vertex at the point [tex]\left(-\dfrac{3}{2},-\dfrac{53}{4}\right).[/tex]
Therefore, the axis of symmetry is given by
[tex]x=-\dfrac{3}{2}.[/tex]
Thus, the vertex form of the given function is [tex]y=5\left(x+\dfrac{3}{2}\right)^2-\dfrac{53}{4}[/tex] and its axis of symmetry is [tex]x=-\dfrac{3}{2}.[/tex]
To convert the quadratic function y = 5x² + 15x - 2 into vertex form, completing the square yields y = 5(x + 1.5)² - 13.25, with an axis of symmetry at x = -1.5.
To rewrite the quadratic function y = 5x² + 15x - 2 in vertex form and find the axis of symmetry, follow these steps:
Complete the square
Start with the given quadratic function:
y = 5x² + 15x - 2
To complete the square, first factor out the coefficient of x² (which is 5):
y = 5(x² + 3x) - 2
Now, focus on the expression inside the parentheses, x² + 3x.
To complete the square, need to add and subtract a constant term that will make it a perfect square trinomial.
add (3/2)² = 2.25 and subtract the same value inside the parentheses:
y = 5(x² + 3x + 2.25 - 2.25) - 2
Simplify the expression inside the parentheses
Now, simplify the expression inside the parentheses:
x² + 3x + 2.25 = (x + 1.5)²
So, the equation becomes:
y = 5[(x + 1.5)² - 2.25] - 2
Distribute the 5 and simplify further
y = 5(x + 1.5)² - 5(2.25) - 2
Rewrite in vertex form
y = 5(x + 1.5)² - 11.25 - 2
y = 5(x + 1.5)² - 13.25
Now, the quadratic function is in vertex form:
y = 5(x + 1.5)² - 13.25
Find the axis of symmetry
The axis of symmetry of a parabolic equation in the form y = a(x - h)² + k is given by x = h.
The axis of symmetry is x = -1.5 (opposite sign of the value inside the parentheses).
Therefore, the axis of symmetry for the quadratic function y = 5x² + 15x - 2 is x = -1.5.
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Express the polynomial −g + 3g + 2g2 in standard form and then classify it.
Answers plz?
1.The graph shows the distance a car travels at a constant speed. What is the speed of the car?
2.Write an equation that represents the speed of the car.Question 2 options:
1. Find the product of 2x4(2x2 + 3x + 4).
2x8 + 3x4 + 4x4
4x6 + 6x5 + 8x4
4x4 + 3x5 + 2x6
3x6 + 4x5 + 5x4
2. Find the product of (3x 4)(2x2 + 2x 1)
6x3 + 2x2 - 5x + 4
6x3 + 14x2 - 11x + 4
6x3 - 14x2 - 5x + 4
6x3 - 2x2 - 11x + 4
3. Find the product of (x2 + 4x 3)(2x2 + x + 6).
2x4 + 9x3 + 4x2 + 21x 18
2x4 + 9x3 + 10x2 + 21x 18
2x4 + 7x3 + 4x2 + 21x18
2x4 + 9x3 + 4x2 + 27x 18,
Sally left a $12.24 tip that represented 17% of her bill. What was the original cost of the meal?
Answer:
72 is the og
Step-by-step explanation:
72*.17=12.24
The original cost of the meal is $72.
What is percentage?In mathematics →
a percentage is a number or ratio expressed as a fraction of 100.it is often denoted using the percent sign "%".Given is that Sally left a $12.24 tip that represented 17% of her bill.
Assume that the original cost of the meal is $[x]. So, we can write -
12.24 = 17% of {x}
12.24 = (17/100) × {x}
{x} = (12.24 x 100)/17
{x} = 1224/17
{x} = 72
Therefore, the original cost of the meal eaten by Sally is $72.
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Jamie rode her bike home for 5 blocks before realizing she forgot her math book at school and would need it for her homework. She rode back to school, found her math book, and then rode home.
Which graph best represents the scenario described?
Answer:
Its B
Step-by-step explanation:
I just took the test
Which is a possible number of real roots for a cubic function? Select all that apply.
0
1
2
3
4
what even is this,
According to Okun's law, if the unemployment rate goes from 2% to 5%, what will be the effect on the GDP?
Answer: decreases by 3%
Step-by-step explanation:
The options for number two is
31.5 m^2
63 m^2
84 m^2
126 m^2
Please help! I will give brainliest to the best explained answer.
find the area of the triange with the given vertices A(0,4), B(0,0), C(2,0)
Melissa and Joey enjoyed dinner at a restaurant. They paid the waiter $100.50 plus a 20% tip. How much did each person pay if they split the total bill in half? $60.30 $70.00 $75.10 $100.10
the answer is $60.30 because half of 100.50 is 50.25 and the both would pay 10%. 10% of 100.50 is 10.05.
10.05+50.25= 60.30
In the question it is already given that Melissa and Joey would have to share the total bill along with the tips at the resturant during their dinner. The paid the water an amount of $100.50 and additional 20% tip.Firstly we have to find the amount of the tip and then add with the total amount to calculate the total amount paid by both of them.
Amount of tips paid to the waiter = (20/100) * 100.50 dollars
= 2010/100 dollars
= 20.10 dollars
So Melissa and Joey paid $20.10 as tips to the waiter.
Then the total amount paid by both of them = (100.50 + 20.10) dollars
= 120.60 dollars
Then the amount paid by Melissa and Joey = 120.60/2 dollars
= 60.3 dollars.
So Melissa and Joey will individually pay $60.3 each.
Imagine if you wrapped a rope tightly around the earth at the equator. how much longer would you have to make the rope if you wanted it to be exactly one foot above the surface all the way around?
PLEASE HELP! An organization surveyed a random sample of their employees about their new vacation policy. Out of the employees surveyed, 83% rated the vacation policy as "delightful" with a margin of error of ±2% and a confidence interval of 95%.
What does the margin of error imply?
A.It can be concluded, with 81% confidence, that between 93% and 97% of all employees will rate the vacation policy as "delightful."
B.It can be concluded, with 95% confidence, that at least 81% of all employees will rate the vacation policy as "delightful."
C.It can be concluded, with 95% confidence, that between 81% and 85% of all employees will rate the vacation policy as "delightful."
D.It can be concluded, with 81% confidence, that at least 95% of all employees will rate the vacation policy as "delightful."
Answer:
The answer is option C.
Step-by-step explanation:
Here, according to the given question, margin of error means that the number of people who rated the vacation as 'delightful' ranges between 81% (83-2) and 85% (83+2) of the population at a 95% confidence interval. The range is formed as 81% to 85% as its given that the margin of error is ±2% .
Therefore, only option C is having both the ranges, 81% and 85%. So, option C is the correct answer.
C.It can be concluded, with 95% confidence, that between 81% and 85% of all employees will rate the vacation policy as "delightful."
The length of a rectangle is 4 cm more than 3 times it’s width. If the area of the rectangle is 15 cm^2, find the width. Use quadratic equations.
Ursula and Andre competed against each other in a swimming race. Ursula swam at a rate of 60 meters per minute and Andre swam at a rate of 48 meters per minute. Ursula gave Andre a 1-minute head start.
Final answer:
Ursula, swimming at 60 meters per minute, and given Andre a 1-minute head start who swims at 48 meters per minute, will catch up to Andre 4 minutes after she starts.
Explanation:
Ursula and Andre competed against each other in a swimming race. Ursula swam at a rate of 60 meters per minute and Andre swam at a rate of 48 meters per minute. Ursula gave Andre a 1-minute head start. To determine who wins the race, we need to calculate the time it takes for Ursula to catch up to Andre, given the head start.
Let's denote the time it takes for Ursula to catch up to Andre as t (in minutes). Since Andre has a 1-minute head start, he would have swum 48 meters before Ursula starts. Therefore, the distance Andre swims in t+1 minutes is 48(t+1). Ursula's swimming distance in t minutes is 60t.
To find out when Ursula catches up to Andre, we set these distances equal to each other:
60t = 48(t+1)60t = 48t + 4860t - 48t = 4812t = 48t = 4Thus, Ursula will catch up to Andre 4 minutes after she starts swimming, which is 5 minutes after Andre starts.
One angle of a right triangle measure a°, where cos a° = 3/5. What is sin(90° - a°)? (PLEASE ANSWER!!!)
How many odd integers are there between 101 and 199?
Answer:
49 odd numbers are there between 101 and 199.
Step-by-step explanation:
Given : 101 and 199.
To find : How many odd integers are there between 101 and 199.
Solution : We have given 101 and 199.
Total number(N) between 101 and 199
N = 199 - 101 = 98
N = 98 .
We can see 98 is even number so, there are half numbers are odd and half numbers are even .
Odd number = [tex]\frac{N}{2}[/tex].
Odd number = [tex]\frac{98}{2}[/tex].
Odd number = 49.
Therefore, 49 odd numbers are there between 101 and 199.
**PLEASE HELP**
The area of a rectangle is 3x^2 + 14x + 8, and the length is x + 4. Which expression best describes the rectangle's width?
A.) 2x + 2
B.) 3x - 2
C.) 3x + 2
D.) 2x + 4
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Please give answer and explain how to solve.
Use the Law of Sines to find the missing angle of the triangle. Find measure of angle b given that c=83 , a=44 and that measure of angle a =31 . A=76.3 degrees B= 15.8 degrees C= 72.7 degrees,
Answer:
C : angle B is 72.7°
Step-by-step explanation:
I did the same Assingnment
Which symbol can be placed in the box to make this number sentence true?64 (13 – 5) + 8 = 16
a.–
b.÷
c.+
d.×
what is the approximate area of a rectangle that is 5ft in length and 15 feet in width
The histogram shows the number of hours volunteers worked one week.
What percent of the volunteers worked 8 to 11 hours or 16 to 19 hours?
Enter your answer in the box.
%
Use the data to create a scatter plot.
Stretch distance
(cm)
Weight
(Newtons)
2.0
1
3.5
2
4.0
2
4.5
3
5.5
5
6.0
6
7.5
8
9.0
9
Weight vs. Stretch Distance
Point
Move
Undo
Redo
Reset
Stretch distance (cm)
Weight (N)
To create a scatter plot for the given data, label the axes with 'Stretch distance (cm)' and 'Weight (Newtons)', plot the points, and observe the correlation. A positive trend is expected as weight increases with stretch distance.
Explanation:To create a scatter plot with the given data of stretch distance versus weight, you would typically use graphing software or graph paper. Here's a step-by-step explanation on how to plot the data:
Label the x-axis as 'Stretch distance (cm)' and the y-axis as 'Weight (Newtons)'.Mark even increments on both axes according to your data range. For stretch distance, you might use increments of 1 cm, and for weight, you could use increments of 1 Newton.Plot each pair of values from the table onto the graph. For example, you would place a point at (2.0 cm, 1 N), (3.5 cm, 2 N), and so on.Once all points are plotted, you may observe a pattern or trend which can be indicated by drawing a line of best fit if appropriate.
After plotting all the points, your scatter plot might show a positive correlation between stretch distance and weight, indicating that weight increases as the stretch distance increases.
PLEASE HELP ASAAAPPPPPP
How could you increase the mass of a wooden block?
Take it to Jupiter where there is more gravity.
Take it to the moon where there is less gravity.
Cutting it in half.
None of the above.,