SOLUTION: It is given that the base and its two lateral side are 10cm in length. Find the measurement of the upper side so that the gutter can the hold the maximum amount of water. Give a re

Question 1141515: It is given that the base and its two lateral side are 10cm in length. Find the measurement of the upper side so that the gutter can the hold the maximum amount of water. Give a reason to support your answer.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Let x be the angle the sides make with a perpendicular to the bases -- e.g., if the angles the sides make with the bottom base are 100 degrees, then x is 10 degrees. Then

the bottom base is 10
the top base is 10+2*10sin(x)
the height (depth) is 10cos(x)

The maximum amount of water is when the cross sectional area is maximum.

The area is height times the average of the two bases.

To find the maximum area, find where the derivative is zero. Note we can ignore the constant 100 and find the maximum value of

By the product rule, the derivative is...

Using

The derivative is zero when cos(2x) = sin(x).

Knowing that the angle x is acute, and that for acute angles sin(x) = cos(90-x), we can determine that the derivative is zero when x is 30 degrees.

This can be confirmed by graphing the area function on a graphing calculator and finding the value of x that produces the maximum area.

ANSWER: The gutter will hold the maximum amount of water when the length of the top base is 10+20*sin(30) = 10+20(1/2) = 10+10 = 20.

RELATED QUESTIONS
Hi, Ok so I have this problem, and it says their are two polygons that are similar. Find (answered by josgarithmetic,KMST)

The lateral surface area of a right square pyramid at the right in the figure above is... (answered by greenestamps)

A regular hexagonal pyramid has one lateral edge which measures 10 cm. The length of one... (answered by KMST)

How do you find the length of one side in a right triangle when given the length of one... (answered by Alan3354)

Find the length of each side of the rectangle so that it has the given perimeter.... (answered by stanbon)

A triangle has two angles and one side length given . The two angles are 62 and 58. The... (answered by Alan3354)

What is the total and lateral surface area of a material whose volume is 54 cubic in.... (answered by ankor@dixie-net.com)

the two longer sides of a right triangle are 8cm and 10cm, What is the length of the... (answered by ewatrrr)

7. P is the centre of the upper face of the rectangular block shown in the figure.... (answered by AnlytcPhil)