Answer to Question #145878 in Analytic Geometry for rem

Question

1. A farmer wants to dig a circular reservoir for water storage. He decided to have it 10- feet wide in diameter. Is there a need for a mango tree that is located at coordinates (-4, 2) from the center of the reservoir to be removed? Support your answer.

 

2. Satellite dishes are usually shaped like a paraboloid, where each cross section is a parabola. This shape of the dish allows radio signals coming in parallel to the axis of the dish to bounce off the surface and meet at the focus. Hence, the receiver should be placed at the focus for a good signal reception. How far should the receiver be from the vertex if the satellite dish is 14 ft across and 5.2 ft deep at the vertex?

 

 

Answer 1

The first task.

There is a need to remove a tree because a distance of a tree from the center of reservoir

is less then a radius of reservoir.

Radius of reservoir is $\frac{10}{2}=5$ feet.

Distance of a tree from the center is $\sqrt{4^2+(-2)^2}=\sqrt{24}$

$5^2>24$ and $5>\sqrt{24}$ .

The second task.

The cross section of the dish is parabola $y=ax^2$ , where vertex is

point (0; 0). On the edge of the dish $y=5.2$ and $x=\frac{14}{2}=7$ .

So we can find $a=\frac{5.2}{7^2}.$ Focus equals

$F=\frac{1}{4a}=\frac{7^2}{4*5.2}=2.36$ feet