Question

A searchlight is shaped like a paraboloid of revolution. If the light source is located 2 feet from the base along the axis of symmetry and the opening is 12 feet​ across, how deep should the searchlight​ be?

Answer 1

Answer:

3 ft.

Step-by-step explanation:

We have that the parabola formula has the following general equation:

x ^ 2 = 4py

p would come being the distension of the vertex until the focus.

They tell us that this value is 3, that is to say p = 3. Replacing we have:

x ^ 2 = 4 * 3 * y

x ^ 2 = 12 * y

reorganizing for and, we are left with:

y = (1/12) * x ^ 2

We know that the opening is 12 feet wide, therefore, the parabola, being symmetry, must be distributed in the same way, in two equal parts, that is, 12/2 = 6. Therefore, being vertical on the x axis must take values of (-6, y), (6, y).

Which means that the value of x is 6 or -6. (it doesn't matter which of the two is when it is squared)

Replacing in the equation obtained:

y = (1/12) * (6 ^ 2)

y = 3

Therefore the depth is equal to 3 ft.

Answer 2

Using the equation of the form of a parabola which can be used to obtain the depth value on the y-axis of a parabola, the depth of the searchlight is 3 feets.

Distance from vertex to focus, p = 3

Using the relation :

x² = 4py

Since p = 3

• Depth of searchlight = y

Substitute the value into the equation :

x² = 4 × 3 × y

x² = 12y - - - - - - - - (1)

From the diagram attached, the intersection made by the opening of the searchlight.

Using either of the coordinates (6, y) or (-6, y) ;

Substitute the value of x = 6 in (1)

x² = 12y

6² = 12y

36 = 12y

Divide both sides by 12

36/12 = 12y/12

y = 3

Therefore, searchlight has a depth of 3 feets.

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