Question

Okay - so where did I mess up?

Answer 1

Answer: x = 105.1

Explanation:

Consider two triangles:

• small Δ (60°) has an opposite side of 91 and an adjacent side of "y"
• large Δ (30°) has an opposite side of 91 and an adjacent side of "x + y"

small Δ:

    tan 60° = $\frac{91}{y}$

⇒ y tan 60° = 91

⇒ y = $\frac{91}{tan60}$

large Δ:

    tan 30° = $\frac{91}{x+y}$

⇒ (x+y) tan 30° = 91

⇒ x + y = $\frac{91}{tan30}$

⇒ y = $\frac{91}{tan30}$ - x

Solve the System:

small Δ: y = $\frac{91}{tan60}$

large Δ: y = $\frac{91}{tan30}$ - x

⇒ $\frac{91}{tan60}$ = $\frac{91}{tan30}$ - x

⇒ $\frac{91}{tan60}$ - $\frac{91}{tan30}$ = -x

⇒ 52.5 - 157.6 = -x

⇒ -105.1 = -x

⇒ 105.1 = x

Your error was using cos (adjacent/hypotenuse) instead of tan (opposite/adjacent).

Answer 2

Check out the attached image below. I added a variable y to help find x

tan(angle) = opposite/adjacent

Focusing on the smaller 30-60-90 triangle, we see that

tan(60) = 91/y

y*tan(60) = 91

y = 91/tan(60)

y = 52.53887

now focus on the largest triangle and we see that

tan(30) = 91/(x+y)

0.57735 = 91/(x+52.53887)

0.57735*(x+52.53887) = 91

0.57735*x + 0.57735*52.53887 = 91

0.57735*x + 30.3333165945 = 91

0.57735*x + 30.3333165945 - 30.3333165945 = 91 - 30.3333165945

0.57735*x = 60.6666834055

x = 60.6666834055/0.57735

x = 105.07782697757

x = 105.1

So the final answer is approximately 105.1