The length of a rectangle is 4 times its width x. If the rectangle is inscribed in a circle, how do you determine the area of the circle as a function of x?

Question

Genesis Allen · Accepted Answer

#17/4pix^2#

The width of the rectangle is #x# and the length is #4# times the width.

Width#=x#
Length#=4x#

When a quadrilateral is inscribed in a circle, the centre of the quadrilateral falls exactly on the centre of the circle.

As you can see, the radius of the circle is equivalent to half the length of the diagonal of the rectangle.

#r=1/2xxsqrt((4x)^2+(x)^2)#
#color(white)(r)=sqrt(17x^2)/2#

Now that we have the radius, let's find the area of the circle using the formula of the area of a circle: #pir^2#.

#A=pi(sqrt(17x^2)/2)^2#
#color(white)(A)=pi((17x^2)/4)#
#color(white)(A)=17/4pix^2#

The area of the circle is #17/4pix^2#.

Nathan Axel · Answer

See explanation.

If a rectangle is inscribed in a circle, then its diameter is the diagonal of the rectangle. If we use the given information and the Pythagorean theorem we get: The circle radius is half of its diameter, so: To calculate the area we use: #A=pir^2# The area is then:

Abigail Allen · Answer

#pi xx 17/4x^2# Firstly, find the length of half a diagonal across the rectangle. The length of a diagonal can be found using pythagorus: #sqrt(x^2 + (4x)^2) = sqrt(17x^2)# #sqrt(17x^2) = sqrt17x# Then you half this length #(sqrt17x)/2# This is the radius of the circle. Therefore as the area of a circle is #pixxr^2#, you can substitute in your value for r as follows: #pi xx ((sqrt17x)/2)^2 = pi xx 17/4x^2#

Nathan Axel · Answer

undefined The area of the circle inscribed by a rectangle is determined by the diagonal of the rectangle, which is the diameter of the circle. The diagonal of the rectangle can be found using the Pythagorean theorem. Once the diagonal is known, the radius (half the diameter) of the circle can be found, and the area of the circle can be calculated using the formula for the area of a circle, A = πr². Since the rectangle's length is 4 times its width, the length is 4x and the width is x. Therefore, the diagonal of the rectangle is √(4x)² + x² = √(16x² + x²) = √(17x²). The radius of the circle is half the diagonal, so r = (1/2)√(17x²) = √(17)x/2. Finally, the area of the circle as a function of x is A(x) = π(√(17)x/2)² = (π/4) * 17x².