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A cone-shaped paper drinking cup is to be made to hold 27 cm of water. Find the height h and radius r of the cup that will use t

andreyandreev [35.5K]

Answer:

r = 2.632 cm
h = 3.722 cm

Step-by-step explanation:

The formula for the volume of a cone of radius r and height h is ...

V = (1/3)πr²h

Then r² can be found in terms of h and V as ...

r² = 3V/(πh)

The lateral surface area of the cone is ...

A = (1/2)(2πr)√(r² +h²) = πr√(r² +h²)

The square of the area is ...

T = A² = π²r²(r² +h²)

Substituting for r² using the expression above, we have ...

T = π²(3V/(πh))((3V/(πh) +h²) = 9V²/h² +3πVh

We want to find the minimum, which we can do by setting the derivative to zero.

dT/dh = -18V²/h³ +3πV

This will be zero when ...

3πV = 18V²/h³

h³ = 6V/π . . . . . multiply by h³/(3πV)

For V = 27 cm³, the value of h that minimizes paper area is ...

h = 3∛(6/π) ≈ 3.7221029

The corresponding value of r is ...

r = √(3V/(πh)) = 9/√(π·h) ≈ 2.6319242

The optimal radius is 2.632 cm; the optimal height is 3.722 cm.

_____

The second derivative test applied to T finds that T is always concave upward, so the value we found is a minimum.

__

Interestingly, the ratio of h to r is √2.

8 0

3 years ago

-3/v=-6 simply as much as possible

Aleksandr [31]

Hey!

In order to simplify this equation, we'll first have to multiply both sides of the equation by v. This will give us v on its own.

Original Equation :
$\frac{-3}{v} = -6$

New Equation {Added Multiply Both Sides by V} :
$\frac{-3}{v} v=-6v$

Solution {New Equation Solved} :
$-3 = -6v$

Now we'll switch sides to get v on the left side of the equation which is generally where we always want the variables to be located in these types of equations.

Old Equation :
$-3 = -6v$

New Equation {Switched} :
$-6v=-3$

Now we'll divide both sides by v to get v on its own.

Old Equation :
$-6v = -3$

New Equation {Added Divide Both Sides by V} :
$\frac{-6v}{v} = \frac{-3}{v}$

Solution {New Equation Solved} :
$v = \frac{1}{2}$

So, this means that in the equation $\frac{-3}{v} =-6$ , $v = \frac{1}{2}$ .

Hope this helps!

- Lindsey Frazier ♥

3 0

3 years ago

Shellie spent $10.56 on oranges that cost $0.88 per orange. How many oranges did she purchase?

vlada-n [284]

You have to do 10.56 plus 88 and you will have your answer

6 0

3 years ago

880,005 rounded to the nearest hundred thousand is

MrMuchimi

Answer:

900,000

Step-by-step explanation:

It's so easy. Don't need to explain.

4 0

3 years ago

Read 2 more answers

What number should be placed in the box to help complete the division calculation? can someone please do the problem (8587/12) s

disa [49]

Answer:

8587 should be in the box

Step-by-step explanation:

we would see it question as: 12/- 8587

12 goes into 85 7 times, so 7

12 goes into 18 1 time, so 1

12 goes into 67 5 times, so 5

then the remainder is 7

or you further divide to get . 5 8 3333333...

so your answer should be 715r7 or 715.58333

5 0

3 years ago