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Difficulty: MediumRight Triangle Trigonometry (SOHCAHTOA)

A straight ramp connects a driveway to a loading dock that is 1010 feet above the ground. The ramp forms an angle θ\theta with the flat ground such that tan⁡(θ)=512\tan(\theta) = \frac{5}{12}. What is the length, in feet, of the ramp?

  1. A
    1414
  2. B
    2424
  3. 2626Answer
  4. D
    2612\sqrt{61}
  5. E
    3030

Answer

The length of the ramp is 2626 feet.
In the right triangle formed by the ground, the loading dock, and the ramp, the dock height (1010 feet) is opposite angle θ\theta, and the ramp is the hypotenuse. Since tan⁡(θ)=oppositeadjacent=512\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12}, setting 10adjacent=512\frac{10}{\text{adjacent}} = \frac{5}{12} yields an adjacent side of 2424 feet. Applying the Pythagorean theorem to find the hypotenuse gives 102+242=676=26\sqrt{10^2 + 24^2} = \sqrt{676} = 26 feet.

Step-by-Step Solution

1
Identify the given information and trigonometric ratio.
The vertical leg opposite angle θ\theta is 1010 feet. The ratio is tan⁡(θ)=oppositeadjacent=512\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12}.
Tangent is defined as the ratio of the side opposite the angle to the side adjacent to the angle in a right triangle.
2
Solve for the length of the adjacent leg (ground distance).
10adjacent=512  ⟹  adjacent=10×125=24\frac{10}{\text{adjacent}} = \frac{5}{12} \implies \text{adjacent} = \frac{10 \times 12}{5} = 24 feet.
Cross-multiplying gives the length of the ground leg.
3
Calculate the length of the ramp (hypotenuse) using the Pythagorean theorem.
\text{ramp length} = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26\text{ feet}.
The ramp forms the hypotenuse of the right triangle, so its length is c=a2+b2c = \sqrt{a^2 + b^2}.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA) and Pythagorean Theorem
Estimated Time:1m 15s
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Topics

Right Triangle Trigonometry (SOHCAHTOA)

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