Question

Hey looking for some help!! I cant seem to get this question, Ive tried for hours and would really appreciate some help! A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 23 feet? I know its optimization, and I have tried all the things with x, y pi/2 deravitives etc, and i cant get it. If someone can help thanks a lot!

Answer 1

Let x be the width of the rectangle and y be its length.
The radius of the semi-circle will be x/2

Perimeter = x + y + y + pi*(x/2) = x(1+pi/2) + 2y
x(1+pi/2) + 2y = 23 (1)

Area A = xy + 1/2 * pi(x/2)^2 (2)

Find y from (1), put it in (2)
To maximize A, find the derivative with respect to x, set it to 0 and solve for x.

Answer 2

doesnt work

Answer 3

Not sure but that is not correct

Answer 4

Are there any accompanying instructions such as how many decimal places or whether leave pi in the answer, etc.?

Answer 5

Nothing else really, a bunch have people have tried tho

Answer 6

I got rid of the pi and simplified all constants to decimals and now I am getting 37.04 square feet. Rounded to 37 ft^2

Answer 7

Equation 1) simplified to y = 11.5 - 1.2854x
Equation 2) simplified to A = xy + 0.3927x^2

Substituting (1) in (2) and simplifying gives A = 11.5x - 0.8927x^2 --- (3)
dA/dx = 11.5 - 1.7854x = 0
x = 6.44 ft.
put x in (3) and A = 37.04 sq. ft.

Answer 8

Wow thanks a lot! Im gonna actually try to learn it of course as well cause I will need to

Answer 9

The other question I cant seem to get is The point on the line −2x+3y+10=0 which is closest to the point (5 , -4) has x coodinate

I thought it would be simple just like plugging it into the distance formula, expressing x in terms of y and plugging. However it doesnt work

Answer 10

You can do this without calculus. If you drop a perpendicular line from the point (5,-4) to the line −2x+3y+10=0, the point where the perpendicular line meets the line will be the closest point.
Find the slope m of the line: −2x+3y+10=0.
The slope of the perpendicular line will be the negative reciprocal: -1/m
Knowing the slope of the perpendicular line and knowing it passes through (5,-4) you can find its equation.
Find where the two lines intersect by solving the two line equations simultaneously.
I am getting x = 3.15.

Do it using calculus:
Distance D between a point (x,y) on the line and (5, -4) is:
D = { (x-5)^2 + (y+4)^2 }^(1/2) --- (2)
Since (x,y) lies on the line it satisfies the equation:
−2x+3y+10=0 --- (1)

They are interested in the x-value. So eliminate y.
Find y from (1), put it in (2)

To minimize the distance, find the derivative D' and set it to zero. Solve for x.
I am getting the same x = 3.15