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

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> 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      Log On


   

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) About Me  (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.

A+=+10cos%28x%29%28%2810%2B%2810%2B20sin%28x%29%29%29%2F2%29

A+=+100cos%28x%29%281%2Bsin%28x%29%29

To find the maximum area, find where the derivative is zero. Note we can ignore the constant 100 and find the maximum value of cos%28x%29%281%2Bsin%28x%29%29

By the product rule, the derivative is...

%28cos%28x%29%2Acos%28x%29%29%2B%28-sin%28x%29%29%281%2Bsin%28x%29%29
cos%28x%29%5E2-sin%28x%29-sin%28x%29%5E2

Using cos%28x%29%5E2-sin%28x%29%5E2+=+cos%282x%29%2C%3Cbr%3E%0D%0A%7B%7B%7Bcos%282x%29-sin%28x%29

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.