The width of a rectangle is 75% the length of the rectangle. the perimeter of the rectangle is 42 inches. a circle is drawn that contains all four vertices of this rectangle. if the area of the circle is xπ square inches, then what is the value of x ?

  x = 56.25

Step-by-step explanation:

The sides of the rectangle are in the ratio 3:4, so those together with the diagonal of the rectangle make a 3-4-5 right triangle. The sum of these adjacent-side ratio units is 3+4 = 7, so the perimeter in ratio units is 2×7 = 14. The actual perimeter is 42 inches, so each ratio unit must stand for ...

  (42 in)/(14 units) = 3 in/unit

The diagonal of the rectangle is the diameter of the circle. Its length is 5 units, or ...

  (5 units)×(3 in/unit) = 15 inches.

Then the radius is half the diameter, or 7.5 inches.

The area of a circle is given by ...

  A = πr² = π(7.5 in)² = 56.25π in²

The value of x is 56.25.