SOLUTION: Find the dimensions and the area of the largest rectangle that can be fit under the graph of y = sin(x), 0 <= x <= π, if one side of the rectangle lies on the positive x-axis.

Algebra ->  Numeric Fractions Calculators, Lesson and Practice -> SOLUTION: Find the dimensions and the area of the largest rectangle that can be fit under the graph of y = sin(x), 0 <= x <= π, if one side of the rectangle lies on the positive x-axis.      Log On


   

Question 1185013: Find the dimensions and the area of the largest rectangle that can be fit under the graph of y = sin(x), 0 <= x <= π, if one side of the rectangle lies on the positive x-axis.
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!


the area of the largest rectangle:
A=L%2AW
L=pi-2x
W=sin%28x%29
A=%28pi-2x%29%2Asin%28x%29

to maximize, take derivative
A'=%28pi-2x%29%2Acos%28x%29-2sin%28x%29
equal it to zero
%28pi-2x%29%2Acos%28x%29-2sin%28x%29=0
x ≈ 0.710462737775517


then
L=pi-2%2A0.710462737775517=1.720667178038759
W=sin%280.710462737775517%29=0.652184623909187
A=1.720667178038759%2A0.652184623909187
A+≈ 1.12 square units

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The height of the rectangle will be y=sin(x); by symmetry, to length of the rectangle will be (pi-2x). So the area of the rectangle will be

A=%28pi-2x%29sin%28x%29

We could try to solve the problem by finding where the derivative of the area function is zero; however, the equation we end up with can't be solved by purely algebraic methods.

So we might as well find the answer by using our graphing calculator or similar tool to find the maximum value of the area function.

ANSWERS: (to a few decimal places)
x=0.7104613
height sin(x) = 0.652184
length pi-2x = 1.72067
area (pi-2x)*sin(x) = 1.122192