Question

A circle plaza measuring 20 meters radius is surrounded by a path of uniform width x m. The area of a path is equal to the area of circle plaza. (Area of circle: πr^2)
(a) Find the width x.
(b) Calculate the cost of paving the path with marbles at $800 per m^2.

Answer 1

Answer:

width x=8.28427 m

Cost=$1005440

Step-by-step explanation:

Area of the plaza is given as;

radius=20 m

pi=3.142

$A_1=\pi *r^2\\A_1=3.142*20^2\\A_1=1256.8m^2$

For the larger circle:

Radius=$(20+x)$m

pi=3.142

$A_2=\pi *r^2\\A_2=3.142*(20+x)^2\\A_2=3.142(x^{2}+40x+400)\\A_2=3.142x^2+125.68x+1256.8$

Area of the path can be computed as the difference of the two circles;

$A=A_2-A_1\\A=3.142x^2+125.68x+1256.8-1256.8\\A=(3.142x^2+125.68x)m^2$

But the area of the path is equal to the area of the plaza. therefore;

$3.142x^2+125.68x=1256.8$

In standard form;

$3.142x^2+125.68x-1256.8=0$

dividing through by 3.142 we get$x^2+40x-400=0$

We for x using the quadratic formula;

$x=\frac{+-b-\sqrt{b^2-4ac} }{2a}$

In our quadratic equation ;

a=1

b=40

c=-400

therefore

$x=\frac{-b+or-\sqrt{b^2-4ac} }{2a} \\x=\frac{-40+or-\sqrt{40^2-4*1*-400} }{2*1}\\ x=\frac{-40+or-\sqrt{3200} }{2}\\ x=\frac{-40+or-56.5685}{2}\\ x=8.28427m\\and\\x=-48.28m$

But distance is non-negative, so

the width x=8.28427m

Cost of paving the path is given as;

Cost=Area*cost per m^2

but the area of the path is equal to that of the plaza, therefore,

$Cost=1256.8*800\\Cost= 1005440$