An architect wants to design a window in the shape of a rectangle capped by a semicircle. If the perimeter of the window is constrained to be 24 feet, what dimensions should the architect choose for the window in order to admit the greatest amount of light?
If the width is x and the rectangle's height is y, then
x+2y + π/2 x = 24
so, y = (24-(1+π/2)x)/2
The area is
a = xy + π/2 (x/2)^2
= x(24-(1+π/2)x)/2 + π/8 x^2
= 12x - (1/2 + π/8)x^2
Now just set da/dx=0 to find where the max occurs. Or, use what you know about the vertex of a parabola.
Study
To find the dimensions that will admit the greatest amount of light, we need to find the dimensions that maximize the area of the window. Let's proceed step-by-step to solve the problem:
Step 1: Identify the variables:
Let's assume the width of the rectangle is "w" and the radius of the semicircle is "r".
Step 2: Formulate the equation for the perimeter:
The perimeter of the window is the sum of the lengths of all sides. For this window, the perimeter can be written as:
Perimeter = 2w + 2r + πr
Given that the perimeter is constrained to be 24 feet, we can write the equation as:
2w + 2r + πr = 24
Step 3: Formulate the equation for the area:
The area of the window is the sum of the area of the rectangle and the area of the semicircle. The area can be written as:
Area = w * (r + w) + (1/2)πr^2
Step 4: Simplify the area equation:
Using the formula for the area of a semicircle (1/2πr^2) and substituting it into the equation, we get:
Area = w * (r + w) + (1/2)πr^2
= wr + w^2 + (1/2)πr^2
Step 5: Rearrange the perimeter equation:
Solve the perimeter equation for "w" by rearranging the terms:
2w + 2r + πr = 24
2w = 24 - 2r - πr
w = (24 - 2r - πr)/2
w = (12 - r(2 + π))/2
w = 6 - (r(2 + π))/2
Step 6: Substitute the value of "w" into the area equation:
Substitute the expression for "w" from step 5 into the area equation from step 4:
Area = wr + w^2 + (1/2)πr^2
= [(6 - (r(2 + π))/2)r] + [(6 - (r(2 + π))/2)^2] + (1/2)πr^2
Step 7: Determine the maximum area:
To find the maximum area, we need to take the derivative of the area equation with respect to "r" and set it equal to zero:
d(Area)/d(r) = 0
Solve this equation to find the value of "r" that maximizes the area.
Step 8: Calculate the dimensions:
Using the value of "r" obtained from step 7, substitute it back into the equation for "w" from step 5 to find the corresponding value for "w". This will give you the dimensions that maximize the area of the window and admit the greatest amount of light.
Note: The exact calculations depend on the specific value of "r" obtained in step 7.
To find the dimensions of the window that will admit the greatest amount of light, let's break down the problem and formulate a mathematical solution.
Let's assume the width of the rectangle is 'w' and the radius of the semicircle is 'r'. Since the perimeter of the window is constrained to be 24 feet, we can set up an equation:
Perimeter of the window = Perimeter of rectangle + Circumference of semicircle
Solving for the perimeter equation:
2w + πr + 2r = 24
Now, we need to express the area of the window in terms of 'w' and 'r'. The area consists of the rectangle and the semicircle:
Area of window = Area of rectangle + Area of semicircle
Area of rectangle = length × width = w × w = w²
Area of semicircle = (1/2) × π × r²
Area of window = w² + (1/2) × π × r²
To maximize the amount of light, we need to find the maximum value of the area. So, we need to maximize the expression w² + (1/2) × π × r².
To find the dimensions that maximize the area, we can take the derivative and set it equal to zero:
d(Area)/dw = 2w + 0 = 2w
d(Area)/dr = (1/2) × π × 2r = πr
Setting the derivatives equal to zero:
2w = 0
πr = 0
Solving for the critical points:
w = 0 (impossible, as width cannot be zero)
r = 0 (impossible, as radius cannot be zero)
Since there are no critical points, we can conclude that the maximum area occurs at the boundaries. In this case, both width and radius must be positive, so we can set up some equalities using the perimeter equation:
2w + πr + 2r = 24
w + r = 12 - (π/2)r
Substituting (π/2)r for w in the area equation:
Area of window = w² + (1/2) × π × r²
Area of window = [(12 - (π/2)r)²] + (1/2) × π × r²
Expanding and simplifying the equation:
Area of window = 144 - 12πr + (π/4)r² + (1/2)πr²
To find the dimensions that maximize the area, we can differentiate this expression with respect to 'r' and set it equal to zero:
d(Area)/dr = 0
By solving this equation, we can find the value of 'r'. Substituting the value of 'r' back into the perimeter equation will give us the value of 'w', and we'll have the dimensions of the window that admit the greatest amount of light.
Once you have the dimensions, you can plug them back into the area equation to find the maximum amount of light the window will admit.