A light bulb is designed by revolving the graph of:

[tex]y = \frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}[/tex]

Over the interval 0 ≤ x ≤ 1/3 about the x-axis, where x and y are measured in feet. Find the surface area of the bulb and use the result to approximate the amount of glass needed to make the bulb. (Glass is 0.015 inch thick).

Answers

Answer 1

Answer:

[tex]\displaystyle 0.251327 \ in. \ of \ glass[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right

Algebra I

Terms/CoefficientsExpand by FOIL (First Outside Inside Last)Factoring

Calculus

Differentiation

Derivative Notation

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Integration

Integration Property: [tex]\displaystyle \int\limits^a_b {cf(x)} \, dx = c \int\limits^a_b {f(x)} \, dx[/tex]Fundamental Theorem of Calculus: [tex]\displaystyle \int\limits^a_b {f(x)} \, dx = F(b) - F(a)[/tex]Area between Two CurvesVolumes of RevolutionArc Length Formula: [tex]\displaystyle AL = \int\limits^a_b {\sqrt{1+ [f'(x)]^2}} \, dx[/tex]Surface Area Formula: [tex]\displaystyle SA = 2\pi \int\limits^a_b {f(x) \sqrt{1+ [f'(x)]^2}} \, dx[/tex]

Step-by-step explanation:

Step 1: Define

[tex]\displaystyle y = \frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}\\Interval: [0, \frac{1}{3}][/tex]

Step 2: Differentiate

Basic Power Rule:                    [tex]\displaystyle y' = \frac{1}{2} \cdot \frac{1}{3}x^{\frac{1}{2} - 1} - \frac{3}{2} \cdot x^{\frac{3}{2} - 1}[/tex][Derivative] Simplify:                [tex]\displaystyle y' = \frac{1}{6}x^{\frac{-1}{2}} - \frac{3}{2}x^{\frac{1}{2}}[/tex][Derivative] Simplify:                [tex]\displaystyle y' = \frac{1}{6\sqrt{x}} - \frac{3\sqrt{x}}{2}}[/tex]

Step 3: Integrate Pt. 1

Substitute [Surface Area]:                                                                             [tex]\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{1+ [\frac{1}{6\sqrt{x}} - \frac{3\sqrt{x}}{2}}]^2}} \, dx[/tex][Integral - √Radical] Expand/Add:                                                               [tex]\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{\frac{81x^2+18x+1}{36x}} \, dx[/tex][Integral - √Radical] Factor:                                                                         [tex]\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{\frac{(9x + 1)^2}{36x}} \, dx[/tex][Integral - Simplify]:                                                                                       [tex]\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {-\frac{|9x + 1|(3x - 1)}{18}} \, dx[/tex][Integral] Integration Property:                                                                     [tex]\displaystyle SA = \frac{- \pi}{9} \int\limits^{\frac{1}{3}}_0 {|9x + 1|(3x - 1)} \, dx[/tex]

Step 4: Integrate Pt. 2

[Integral] Define:                                                                                             [tex]\displaystyle \int {|9x + 1|(3x - 1)} \, dx[/tex][Integral] Assumption of Positive/Correction Factors:                                 [tex]\displaystyle \frac{9x + 1}{|9x + 1|} \int {(9x + 1)(3x - 1)} \, dx[/tex][Integral] Expand - FOIL:                                                                                 [tex]\displaystyle \frac{9x + 1}{|9x + 1|} \int {27x^2 - 6x - 1} \, dx[/tex][Integral] Integrate - Basic Power Rule:                                                         [tex]\displaystyle \frac{9x + 1}{|9x + 1|} (9x^3 - 3x^2 - x)[/tex][Expression] Multiply:                                                                                      [tex]\displaystyle \frac{(9x + 1)(9x^3 - 3x^2 - x)}{|9x + 1|}[/tex]

Step 5: Integrate Pt. 3

[Integral] Substitute/Integral - FTC:                                                              [tex]\displaystyle SA = \frac{- \pi}{9} (\frac{(9x + 1)(9x^3 - 3x^2 - x)}{|9x + 1|})|\limits_{0}^{\frac{1}{3}}[/tex][Integrate] Evaluate FTC:                                                                                [tex]\displaystyle SA = \frac{- \pi}{9} (\frac{-1}{3})[/tex][Expression] Multiply:                                                                                     [tex]\displaystyle SA = \frac{\pi}{27} \ ft^2[/tex]

It is in ft² because it is given that our axis are in ft.

Step 6: Find Amount of Glass

Convert ft² to in² and multiply by 0.015 in (given) to find amount of glass.

Convert ft² to in²:                    [tex]\displaystyle \frac{\pi}{27} \ ft^2 \ \div 144 \ in^2/ft^2 = \frac{16 \pi}{3} \ in^2[/tex]Multiply:                                   [tex]\displaystyle \frac{16 \pi}{3} \ in^2 \cdot 0.015 \ in = 0.251327 \ in. \ of \ glass[/tex]

And we have our final answer! Hope this helped on your Calc BC journey!


Related Questions

What is the value of x?
A. 6
B. 5
c. 4
D.3

Answers

Answer:

D

Step-by-step explanation:

CD is an angle bisector and divides the opposite side into segments which are proportional to the other 2 sides, that is

[tex]\frac{BC}{AC}[/tex] = [tex]\frac{BD}{AD}[/tex] , substitute values

[tex]\frac{4}{3}[/tex] = [tex]\frac{x}{2.25}[/tex] ( cross- multiply )

3x = 9 ( divide both sides by 3 )

x = 3 → D

Have to find x,y and z.

Answers

Answer:

z= 2√2

x=2

y=2√2

On Friday, a local restaurant served 150 customers. Of these, 114

people ordered soup. What percent of the customers ordered soup?


A: 13%

B: 66%

C: 34%

D: 76%

Answers

The answer is D, because we will divide 114 by 150 and the total converted to percentage is 76

the number of employees for a certain company had been decreasing each year by 8%. If the company currently has 860 employees and this rate continues, fund the number of employees in 19 years.

Answers

Answer:

approximately 684 employees

Step-by-step explanation:

So this is an exponential equation since there is a dividing decrease, not just subtracting. The standard equation is,

[tex]y = a(b)^{x} [/tex]. [tex]a[/tex] is initial value (860 employees) and [tex]b[/tex] is the change [tex]-8%[/tex] or [tex]0.92[/tex]. The [tex]x[/tex] would be the period of time, [tex]19[/tex]. This equation would be [tex]y = 860(0.92)^{19} [/tex].

After solving, the equation would become the following: [tex]y = 860(0.92)^{19} = 176.387242442≈176[/tex]. After subracting the number of lost employees from orginal employees ([tex]860-176=684[/tex]), we see that the company will have approximately 684 employees left after 19 years.

Five families have signed up to bring a pie. If a single serving of pie is 1/6 of the entire pie, how many people will 5 pies serve? ​

Answers

Answer:

30

Step-by-step explanation:

That means that there are 6 pie pieces in each pie so just multiply by 5

another thing we can do is get rid of 1/6 by multiplying by 6 which means we also have to multiply 5 by 6

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Answers

Answer:

yea its right here

Step-by-step explanation: sike

in the equation x+12=30 does x=16 explain your answer

Answers

no x does NOT equal 16. you would substitute x for 16 which gives you 16+12. That equals 28 not 30. The correct answer is x=18

Noor, Layla, and Diego play a trivia game. Noor’s score is -2. Layla’s score is 3/4 of Noor’s score. Diego’s score is 2/3 of Layla’s score. What is Diego’s score?

Answers

Answer:

layla is savege score in mobile legends and diego's score in ml is 20

find the trigonometric ratio

this is geometry with trig

Answers

Answer:

tanZ = [tex]\frac{20}{15}[/tex]

Step-by-step explanation:

tanZ = [tex]\frac{opposite}{adjacent}[/tex] =  [tex]\frac{XY}{ZY} }[/tex] = [tex]\frac{20}{15}[/tex]

yoo someone help asaapp!​

Answers

Answer:

A  76

Step-by-step explanation:

7x+8+7x-2+5x+16+110=360

19x+132=360

19x=228

x=12

∠E=5(12)+16

    =76

Answer:

76. hope it helps.................



Rewrite in simplest terms: 8(-10r - 9) + 9r



Answers

Answer:

-2

Step-by-step explanation:

you answer the brackets first then add 9 then add 8

On a coordinate plane, the coordinates of vertices R and T for a polygon are R(−7, 3) and T(3, 3). What is the length of Side RT of the polygon?


4 units

6 units

10 units

16 units

Answers

Answer: 6 units
Hope this helps

Answer:

The answer is 10 units because if you add up all the units that would be on the side you will get 10

Step-by-step explanation:

Paul won 37 super bouncy balls playing hoops at county fair. At school he gave two to every student in his math class. He only had 3 remaining. How many students are in his class?

Answers

Answer:

17 students

Step-by-step explanation:

If he started with 37 and ended with 3 we have to subtract those three first so that we know how many we are actually working with so, 37-3=34. 34 balls is the amount distributed among the class. 34/2 is 17.

What is the value of tan(60°)?
A
12
30°
B
60°
14

Answers

Answer:

the answer is 1.7320508076

Elijah is planning a circular flower garden with a 12 foot diameter. He will put plastic edging along the flowerbed. How many feet of edging will Elijah need to enclose his flower garden? Round to the nearest tenth place. (Use 3.14 for π)

Answers

Answer:

[tex]37.7\ \text{foot}[/tex]

Step-by-step explanation:

The length of the edging will be equal to the circumference of the circular flower garden.

Here, the diameter of the circular garden is [tex]d=12\ \text{foot}[/tex]

Circumference of a circle is given by

[tex]C=\pi d\\\Rightarrow C=3.14\times 12\\\Rightarrow C=37.68\approx 37.7\ \text{foot}[/tex]

The length of the edging is [tex]37.7\ \text{foot}[/tex].

Seven people are going to share an 84-ounce bottle of soda. How many ounces will each person get to drink? Choose the correct equation and answer for this situation. I THINK THIS IS ABOUT RASIOS SO PLS HELP

Answers

Answer:

7 ÷ 84 = 84 = 12 ounces

             _

             7

Step-by-step explanation:

The equation will be 7x = 84. Then the number of ounces will each person get to drink will be 12.

What is the linear system?

A linear system is one in which the parameter in the equation has a degree of one. It might have one, two, or even more variables.

Seven people are going to share an 84-ounce bottle of soda.

Let x be the number of the ounces bottle of soda of each person.

Then we have the equation will be

7x = 84

 x = 84 / 7

 x = 12

Then the number of ounces will each person get to drink will be 12.

More about the linear system link is given below.

https://brainly.com/question/20379472

#SPJ2

If a line passes through (4,13) and (6,19) what is the slope? And can you please explain how you got your answer

Answers

Answer:

The slope is 3.

Step-by-step explanation:

The equation for slope is either [tex]\frac{y2-y1}{x2-x1}[/tex] or [tex]\frac{y1-y2}{x1-x2}[/tex]. So this should be [tex]\frac{19-13}{6-4}[/tex], which is 3.

Answer:

Slope =

6

2

Step-by-step explanation:

First find the rise over run

Second use the format

Y2-y1

———-

X2-x1

Using the format you will get this

19-13

——

6-4

Next solve

Which then you get

6

2

hi can you explain base numbers?​

Answers

A number base is the number of digits or combination of digits that a system of counting uses to represent numbers. A base can be any whole number greater than 0. The most commonly used number system is the decimal system, commonly known as base 10. ... Binary is the most commonly used non-base 10 system.

Step-by-step explanation:

hope it helps ☺️

What are the values of x and y?
A.
x equals 136 over 15 and y equals 17 over 15
B.
x equals 64 over 15 and y equals 17 over 15
C.
x equals 8 over 15 and y equals 136 over 15
D.
x equals 64 over 15 and y equals 136 over 15

Answers

Answer:

d. 8 over 15 = 136 over 15 this is the right answer okay

PLZZZ HELP ME OUT........................ PLZZ​

Answers

Answer:

a)  constant of proportionality is 3/5

b)  3.5cm per 1 minute

c)  21 cm water in trough after 6 min

d)  70 cm

Step-by-step explanation:

d)  21 / 6 = x / 20

    7 / 3 = x / 20    

    2x = 140

    x = 70

It would be D 20 minutes water levels

A flying squirrel's nest is 6 meters high in a tree. From its nest, the flying squirrel glides 8 meters to reach an acorn that is on the ground. How far is the acorn from the base of the tree? If necessary, round to the nearest tenth.

Answers

Answer:

2 meters

Step-by-step explanation:

ik you want the points
plzz​

Answers

Answer:

Line 1: Undefined

Line 2: -4/3

Line 3: 4/3

Line 4: Undefined

Line 5: 7/3

Line 6: -7/3

Line 7: 0

Line 8: 0

Line 9: Undefined

Line 11: 0

Line 10 & 12: Undefined

Step-by-step explanation:

Use the slope formula (you can just search up the slope formula)

Also, always remember that a vertical line's slope will be undefined and a horizontal line's slope will be 0 (zero).

I hope this helps!

help for brainlest what can be use ti

Answers

Answer:

2x(5+13)

Step-by-step explanation:

add then multiply by 2

Answer:

to find perimeter

= 2(L+w)

= 2(5+13)

10+26

36

so the answer is 2×(5+13)

hope it's helpful ❤❤❤❤

THANK YOU.

SOMEONE DO THESE QUESTIONS FOR ME ASAP!! - GIVING BRAINLIEST

Answers

Answer:

1 Question= 26

2 Question=41

3 Question= 57

4 Question= 15

5 Question = 20

Step-by-step explanation:

Try the following puzzle:

A bat and ball cost $1.10.

The bat costs one dollar more than the ball.

How much does the ball cost?

Answers

Answer:

10 cents

Step-by-step explanation:

the bat cost 1 dollar so subtract 1 dollar from 1.10 and you are left with .10 or 10 cents

Answer:

the ball costs $0.05

Step-by-step explanation:

if the total is $1.10, and the bat is $1 more than the ball, you do

1+2x=1.10. do the algebra, and you get 0.05. hope this helps.

8t=−32

Help a sister out

Answers

Answer:

t=-4

Step-by-step explanation:

Answer:

t = -4

Step-by-step explanation:

divide each term by 8 and simplify

Let sin(2x)-sin(x)=0, where 0 ≤ x < 2pi. What are the possible solutions for x?

Answers

Step-by-step explanation:

sin2x - sinx = 0

2sinxcosx - sinx = 0

(Double Angle Formula)

sinx(2cosx - 1) = 0

Either sinx = 0 or cosx = 1/2.

When sinx = 0,

x = 0 or x = π.

When cosx = 1/2,

x = π/3 or x = 5π/3.

Hence the solutions are

x = 0, x = π/3, x = π or x = 5π/3.

find ordered pairs: choose two of the answers below
y=2/3x+5

thank you v much//.
a.(-5,-10)
b.(-3,3)
c.(6,9)
d.(2,8)

Answers

Answer:

b -3,3 c 6,9 hope this helps

What is 5+2 x 1-3+8 and 2+5 x 1

Answers

Answer:

Step-by-step explanation:

1) 5+2*1-3+8 = 5+2-3+8 = 7-3+8 = 4+8 = 12

2) 2+5*1 = 2+5 = 7

Answer:

a) 5+2 x 1-3+8

5+2-3+8

=12

b) 2+5 x 1

2+5

=7

Step-by-step explanation:

One less than five times a number is fewer
than twenty-four.

Answers

Answer:

x < 5

Step-by-step explanation:

[tex]5x - 1 < 24\\5x < 25\\x < 5[/tex]

Answer:

any number under 5

Step-by-step explanation:

5(5) -1 =24

so anything under that would equal fewer than 24

Is this what you're asking?

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