A front window on a new home is designed as a rectangle with a semicircle on the top. If the window is designed to let in a maximum amount of light, and the architect fixes the perimeter of the entire window at 600 inches, determine the radius r and rectangular height h so as to maximize the area
area= w*h+1/2 PI (w/2)^2
but 600= w+2h+PI*(w/2)
or h=300-w/2 (PI-1/2)
area= w(above h) + 1/8 PI w^2
take the derivative of area wrespect to w
darea/dw=0=you do it.
To maximize the area of the window, we need to find the dimensions that will maximize the combined area of the rectangle and the semicircle.
Let's assign variables to the important dimensions:
- Width of the rectangle: w
- Height of the rectangle: h
- Radius of the semicircle: r
From the given information, we can write the equation for the perimeter of the entire window:
Perimeter = 2w + 2h + πr + 2r = 600
Since there are two sides of the rectangle that form the base of the semicircle, we include two radii in the equation.
To proceed, we'll need to rewrite the perimeter equation in terms of a single variable.
Let's solve the equation for w in terms of the other variables:
2w = 600 - (2h + πr + 2r)
w = (600 - 2h - πr - 2r) / 2
w = (300 - h - (π/2)r) / 1
Next, we can write the equation for the area of the window:
Area = (Length of rectangle) × (Width of rectangle) + (Area of semicircle)
Area = w * h + (1/2 * πr^2)
Now, we can express the area equation in terms of a single variable, r, using the expression we derived for w:
Area = [(300 - h - (π/2)r) / 1] * h + (1/2 * πr^2)
To maximize the area, we'll differentiate the area equation with respect to r and set it equal to zero:
d(Area)/dr = - (π/2)r - πrh + πr^2 = 0
Let's solve this equation for r:
- (π/2)r - πrh + πr^2 = 0
πr^2 - πr(h + (1/2)) = 0
Factoring out πr:
πr(r - (h + 1/2)) = 0
Since we're looking for a positive radius, we have:
r = h + 1/2
Now, we can substitute this value of r back into the equation for w to find h:
w = (300 - h - (π/2)r) / 1
w = (300 - h - (π/2)(h + 1/2)) / 1
w = (300 - h - (π/2)(h + 1/2)) / 1
w = (300 - h - (π/2)h - π/4) / 1
w = (300 - (π/2 + 1)h - π/4) / 1
w = (300 - (π/2 + 1)h - π/4) / 1
w = (300 - (π/2 + 1)h - π/4) / 1
w = (300 - (πh/2 + h + π/4)) / 1
w = (300 - h(π/2 + 1) - π/4) / 1
Now, we can substitute the value of r back into the perimeter equation to find the value of h:
2w + 2h + πr + 2r = 600
2[(300 - h(π/2 + 1) - π/4) / 1] + 2h + π(h + 1/2) + 2(h + 1/2) = 600
[(300 - h(π/2 + 1) - π/4) / 1] + h + π(h + 1/2) + (h + 1/2) = 300
[(300 - h(π/2 + 1) - π/4) / 1] + h + πh + (π/2) + h + (1/2) = 300
(300 - h(π/2 + 1) - π/4) + h + πh + (π/2) + h + (1/2) = 300
300 - h(π/2 + 1) - π/4 + h + πh + (π/2) + h + (1/2) = 300
300 + h(1 - π/2 - π) - π/4 + (π/2) + (1/2) = 300
h(1 - π/2 - π) - π/4 + (π/2) + (1/2) = 0
h(1 - π/2 - π) = π/4 - (π/2) - (1/2)
h(1 - π/2 - π) = -π/4 - π/2 - 1/2
h = (-π/4 - π/2 - 1/2) / (1 - π/2 - π)
Finally, plug in the value of h to calculate the radius r:
r = h + 1/2
Hence, we have determined the values of h and r that will maximize the area of the window.
To maximize the area of the window, we need to use calculus to find the optimal dimensions. Let's break down the problem and solve it step by step:
1. Define the variables:
- Let r be the radius of the semicircle.
- Let h be the height of the rectangle.
2. Express the perimeter in terms of the variables:
- The perimeter of the entire window is the sum of the semicircle's circumference and the rectangle's perimeter.
- The circumference of the semicircle is given by πr, and the rectangle's perimeter is 2h.
- So, the total perimeter is given by the equation: 2r + 2h + πr = 600.
3. Solve the equation for one variable:
- Rearrange the equation to express one variable in terms of the other.
- 2r + 2h + πr = 600
- 2r + πr = 600 - 2h
- Factor out r: r(2 + π) = 600 - 2h
- Divide both sides by (2 + π): r = (600 - 2h)/(2 + π)
4. Express the area in terms of one variable:
- The area of the entire window is the sum of the rectangle's area and half of the semicircle's area.
- The rectangle's area is given by h * 2r, and the semicircle's area is 0.5 * π * r^2.
- So, the total area is given by the equation: Area = 2rh + 0.5 * π * r^2.
5. Substitute the expression for r into the area equation:
- Area = 2h((600 - 2h)/(2 + π)) + 0.5 * π * ((600 - 2h)/(2 + π))^2.
6. Simplify the area equation:
- Area = h(600 - 2h)/(1 + π) + (0.5π * (600 - 2h)^2)/(4 + 2π).
7. Take the derivative of the area equation:
- To find the maximum area, we differentiate the area equation with respect to h.
- d(Area)/dh = (600 - 4h)/(1 + π) + (2π * (600 - 2h))/(4 + 2π).
8. Set the derivative equal to zero and solve for h:
- Setting the derivative equal to zero gives us (600 - 4h)/(1 + π) + (2π * (600 - 2h))/(4 + 2π) = 0.
- Solve this equation to find the value of h.
9. Substitute the value of h back into the expression for r:
- Use the equation r = (600 - 2h)/(2 + π) to determine the corresponding value of r.
By following these steps, you can find the values of h and r that maximize the area of the window.