Consider a window the shape of which is a rectangle of height h surmounted a triangle having a height T that is 1.3 times the width w of the rectangle.
If the cross-sectional area is A, determine the dimensions of the window which minimize the perimeter.
h=______
w=______
What I did was:
Area:
Rectangle: h*w
Triangle: w * 1.3w / 2 = 0.65w^2
The complete area: hw + 0.3w^2.
Perimeter:
3 sides of the rectangle: 2h+w
Twice the sloped side of the triangle: s^2 = (w/2)^2 + (1.3w)^2 = w^2(0.25+1.69)=1.94w^2, so s = 1.39w.
The complete perimeter p = 2h+w+2.78w = 2h+3.78w
A = hw + 0.65w^2
A - 0.65w^2 = hw
A/w - 0.65w = h
p = p(w) = 2h + 3.78w
= 2(A/w-0.65w) + 3.78w
= 2A/w - 1.3w + 3.78w
= 2A/w + 2.48w
p'(w) = (-1)2A/w^2 + 2.48
p'(w_min) = (-1)2A/w_min^2 + 2.48 = 0
-2A + 2.48w_min^2 = 0
w_min^2 = A/1.24
w_min = sqrt(A/1.24)
So the dimensions I got are:
w_min = sqrt(A/1.24)
h_min = A/w_min - 0.65w_min = A/sqrt(A/1.24) - 0.6sqrt(A/1.24) =
= sqrt(1.24A)-0.65sqrt(A) = sqrt(A) [sqrt(1.24)-0.65].
Both answers are wrong... please help...
To find the dimensions of the window that minimize the perimeter, you correctly derived the equations for the area and the perimeter. However, there seems to be an error in your calculations. Let's go through the solution again to find the correct dimensions.
First, let's define the dimensions of the window:
- Width of the rectangle: w
- Height of the rectangle: h
- Height of the triangle: T = 1.3w
The cross-sectional area of the window is the sum of the areas of the rectangle and the triangle:
A = Area of rectangle + Area of triangle
A = hw + 0.5(T)(w)
A = hw + 0.5(1.3w)(w)
A = hw + 0.65w^2
Next, let's determine the perimeter of the window:
P = Perimeter of rectangle + Perimeter of triangle
P = 2h + 2w + 2(T)
P = 2h + 2w + 2(1.3w)
P = 2h + 2w + 2.6w
P = 2h + 4.6w
Now, we want to minimize the perimeter P with respect to the variable w. To do this, we will take the derivative of P with respect to w and set it equal to zero:
dP/dw = 0
d/dw (2h + 4.6w) = 0
4.6 = 0
This is not possible, which means there is no value of w that minimizes the perimeter.
Therefore, there seems to be an error in the problem statement or the formulation of the window. It is not possible to find the dimensions that minimize the perimeter based on the given information.