Answer the following:

Question

The base of a lifeguard tower is at point A. On a level with A in the same straight line, points B and C were chosen so that the distance between B and C is 12 meters. From points B and C the angles of elevation of the top of the tower are 7⁰ and 5⁰, respectively. Which equation can be used to find the distance, x meters, between points A and B?

Choices:

x = (12 tan 5⁰ tan 7⁰) / (tan 7⁰ - tan 5⁰)

x = (12 tan 5⁰) / (tan 7⁰ - tan 5⁰)

x = 12 tan 5⁰

x = 12 tan 7⁰

Answer 1

The base of a lifeguard tower is at point A. On a level with A in the same straight line, points B and C were chosen so that the distance between B and C is 12 meters. From points B and C the angles of elevation of the top of the tower are 7⁰ and 5⁰, respectively. Which equation can be used to find the distance, x meters, between points A and B?

Choices: a) x = (12 tan 5⁰ tan 7⁰) / (tan 7⁰ - tan 5⁰), b) x = (12 tan 5⁰) / (tan 7⁰ - tan 5⁰), c) x = 12 tan 5⁰, d) x = 12 tan 7⁰

Detailed Solution:

Given/Known: <A = 90⁰, <B = θ = 7⁰, <C = ∝ = 5⁰, CB = 12, AB = x = ??, AC = x + 12, h = Height of tower

* h/AC = h/(x + 12) = tan ∝ ==> h = (x + 12) tan ∝ <== Equation 1

* h/AB = h/x = tan θ ==> h = x tan θ <== Equation 2

Equation 1 = Equation 2

(x + 12) tan ∝ = x tan θ

x tan ∝ + 12 tan ∝ = x tan θ

x tan θ – x tan ∝ = 12 tan ∝

x (tan θ – tan ∝) = 12 tan ∝

x = (12 tan ∝) / (tan θ – tan ∝)

x = (12 tan 5⁰) / (tan 7⁰ – tan 5⁰) <== Exact Solution ( b) x = (12 tan 5⁰) / (tan 7⁰ - tan 5⁰))

x = 29.7446457 <== Approximated Solution