Question

I have a high definition antennae mounted on my house in my backyard. As I don’t exactly have a high ladder, it is mounted quite low, and we don’t receive many channels. The current height is 12 feet.

My friend has offered to let me borrow a 15 foot high ladder. It is recommended to mount the lad- der at a 76 degree angle to the ground to prevent it from sliding out. I will be mounting the antennae 4 feet above the top of the ladder.

A) How far away from the house do I need to put the foot of the ladder?

B) At what height will the top of the ladder touch my house?

C) What height would I be mounting the antennae at?

D) What are the values which describe the triangle above (involving the ladder and the point of con- tact for the ground and the house) for

cos(θ) =?

sin(θ) =?

tan(θ) =?

sec(θ) =?

csc(θ) =?

cot(θ) =?

State each as a fraction involving x, y, r.

Answer 1

Answers:-

(A)x= 3.63 feet

(B)y=14.55 feet

(C) 18.55 feet

(D) Trignometric values in terms of x,y and r are

Solution:-

Let us suppose ladder is placed at x feet distance from the house, the ladder touch at the top of the house at y feet distance and the length of the ladder is r feet.

Let the position of Antennae is Point A which is 4 feet above the top of the house at which ladder touches (point B) and BC is ladder and the bottom of the house is at point D.

(As shown in figure below)

(A)

Now, In ∆BDC <D=90°,<C=76°, r= 15 ft

We know that

On putting the values,we get

Or

Or x≈3.63 feet

Hence, the ladder foot is at 3.63 feet distance from the house.

(B)

Again in the same triangle,

On putting the values,we get

Or y ≈ 14.55 feet

Hence, the ladder touches the antennae at the top of the house at 14.55 feet.

(B)Since Antennae is mounted at 4 get above the top of the ladder.

So, the height at which Antennae is mounted is

=14.55+4

=18.55 feet

(D) The trignometric values are as follows-

Since.

So,

Since,

So,

Since

So,

Since,

So,

Since,

So,

Since,

So,

Hence, these are required values in terms of x ,y and r as defined above.