Question

Please help its due tonight.

A bridge crosses a circular lake. The bridge is represented by the function

y −x = 2 and the lake is represented by the function x^2 +y ^2 = 100.

a. What is the radius of the lake?

b. Find the length of the bridge.

Answer 1

Answer:

a) The radius of the lake to be r=10 units.

b) $14\sqrt{5}$ units

Step-by-step explanation:

The lake has equation: $x^2+y^2=100$

We can rewrite this as $x^2+y^2=10^2$

Comparing this to  $x^2+y^2=r^2$

We have the radius of the lake to be r=10 units.

b)  The bridge is represented by the function  y −x = 2

This is the same as y=x+2

We substitute this into the equation of the circle to get:

$x^2+(x+2)^2=100$

$x^2+x^2+4x+4-100=0$

$2x^2+4x-96=0$

$x^2+2x-48=0$

$(x+8)(x-6)=0$

$x=-8,x=6$

When x=8, y=2(8)+2=18

When x=-6, y=2(-6)+2=-10

The length of the bridge is the distance between the points (8,18) and (-6,-10)

$=\sqrt{(8--6)^2+(18--10)^2}$

$=\sqrt{196+784}$

$=\sqrt{196+784}$

$=\sqrt{980}$

$=14\sqrt{5}$