how to find perimeter of rectangle forming diam's of 2 semi-

[wootn]

Good afternoon,

My daughter and I couldn’t figure out the right answer for the problem below. I got P= 76.5 and my daughter is P= 162. Her teacher said her P= too high and my P= 76.5 is close. Please help us. Thanks,

Problem:
The adjacent sides of a rectangle form diameters of two semicircular regions. The rectangle region has an area of 81. If the ratio of the area of the semicircular regions is 16:1, what it's the perimeter of the rectangle?

 

[Denis]

Re: how to find perimeter of rectangle?

Let width = x, length = y : then xy = 81 ; y = 81/x

circle with diameter x: area = pi (x/2)^2
circle with diameter y: area = pi (y/2)^2
so:
pi (x/2)^2 / pi (y/2)^2 = 1 / 16 ; the pi's cancels out, and:
16(x/2)^2 = (y/2)^2 ; since y = 81/x :
16(x/2)^2 = (81/(2x))^2
16x^2/4 = 6561/(4x^2)
4x^2 = 6561/(4x^2)
16x^4 = 6561
x^4 = 6561/16
x = 4.5 ; y = 81/4.5 = 18

P = 2(4.5) + 2(18) = 45

Hope that helps.

 

[skeeter]

Re: how to find perimeter of rectangle?

if the semicircular areas are in the ratio 16:1, then the linear radii dimensions, and hence the diameters, have a ratio of equal to the square root of 16:1, or 4:1.

so, the two sides of the rectangle can be considered as 4x and x

4x[sup:v9n272qy]2[/sup:v9n272qy] = 81
x = 9/2

perimeter = 10x = 10(9/2) = 45