I'm a bit confused on how to olve this, I keep on getting r = 1.98cm but apparently the answer is 5.91cm? Please help.

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1 Answer
Jan 17, 2018

5.9cm

Explanation:

Using the radii of the circles, we can create an equation for the box using length (#l#) and width (#w#).

#l = r + r + (r - 5) + (r - 5) = 4r - 10#

#w = r + r = 2r#

As the area of a rectangle is length times width (#l*w#), we can find the total area (#A_"rectangle"#) of the rectangle in terms of r:

#A_"rectangle" = l*w #
#A_"rectangle" = (4r - 10) * 2r = 8r^2 - 20r#

The area of the shaded region is #250cm^2#, so the area of the circles (#A_"circles"#) would be the area of the rectangle minus the area of the shaded region:

#A_"circles" = A_"rectangle" - 250#
#A_"circles" = 8r^2 - 20r - 250#

Another way of expressing the area of the circles would be to just use their given radii and the formula #A_"circle" = πr^2#:

#A_"circle1" = πr^2#
#A_"circle2" = π(r-5)^2 = πr^2 - 10πr + 25π#

As #A_"circles"# is just the area of the two circles added together,

#A_"circles" = A_"circle1" + A_"circle2"#
#A_"circles" = πr^2 + (πr^2 - 10πr + 25π)#
#A_"circles" = 2πr^2 - 10πr + 25π#

Now we have two equations for #A_"circles"# in terms of r, which means we can solve for r (aka what the problem is asking for):

#A_"circles" = 8r^2 - 20r - 250#
#A_"circles" = 2πr^2 - 10πr + 25π#

#8r^2 - 20r - 250 = 2πr^2 - 10πr + 25π#
#8r^2 - 20r - 250 - 2πr^2 + 10πr - 25π = 0#
#(8-2π)r^2 +(-20+10π)r + (-250-25π) = 0#

Using the quadratic formula we can solve for r (the values in the parentheses are a, b, and c respectively):

#(-(-20+10π) +- sqrt((-20+10π)^2 - 4(8-2π)(-250-25π)))/(2(8-2π))#

We can "simplify" this to get:

#((20-10π +- sqrt(400 - 400π + 100π^2 + 8000 - 1200π - 200π^2))/(16-4π))#

#((20-10π +- sqrt(8400 - 1600π -100π^2))/(16-4π))#

As you see this is still very complicated... So I would suggest using the magic of the calculator at this point and solving for r. When you do so, you get:

#r = 10.9, - 17.6#

As r can't be negative because the problem is talking about geometry, r must equal 10.9cm.

The radius of the smaller circle is then r - 5 (as shown in the picture):

#r_"small" = r - 5 = 10.9 - 5 = 5.9#