Question

WILL GIVE BRAINLIEST

You must find the horizontal distance between two towers (points A and B) at the same elevation on opposite sides of a wide canyon running east and west. The towers lie directly north and south of each other. You mark off an east/west line CD running perpendicular to AB.



A: From C you measure the angle between the two towers (angle ACB) as 88.60°. Given the distance from C to B is 389 feet, write an equation and solve it to find an expression for the distance AB to the nearest whole foot. (note: AB is perpendicular to CD.)



B: You want to check your work to make sure it’s right.You should be able to both measure and compute the angle at D. Knowing the distance between the two towers from above and the distance BD is 459 feet, what is the angle at D to the nearest hundredth degree?



C: What is angle CAD in radians? Give your answer rounded correctly to 4 decimal places.

Answer 1

A:

We use SOHCAHTOA. We have the value of the adjacent and want the opposite (the angle is ACB). So we use tangent.

$tan(88.6) = \frac{x}{389}$

Solve for x:

$389 \times tan(88.6) = x \\ x = 15916.873$

So, AB = 15917 (to nearest foot).

B:

It's right.

More SOHCAHTOA. We know opposite and adjacent and want to know angle, so more tangent.

$tan(d) = \frac{15917}{459} \\ d = arctan( \frac{15917}{459} ) = 88.348$

Angle of D is 88.35° (nearest hundredth).

C:

Think of this as two triangles: ACB and ABD. Angle BAD is 180 - 90 - 88.35 = 1.65° (since all the angles of a triangle have to add to 180).

Angle CAB is 180 - 90 - 88.6 = 1.4°.

Angle CAD = BAD + CAB.

1.65° + 1.4° = 2.05°.

Multipy by π÷180 to convert to radians:

$2.05 \times \frac{\pi}{180} = 0.0358$

Therefore, CAD = 0.0358 rad.