Problem 46 - Algebra and Trigonometry

Question

A loading ramp 10 feet long that makes an angle of \(18^{\circ}\) with the horizontal is to be replaced by one that makes an angle of \(12^{\circ}\) with the horizontal. How long is the new ramp?

Step-by-Step Solution

Answer

The length of the new ramp is approximately 14.86 feet.

Step 1 - Identify known values

The length of the original ramp is 10 feet. It makes an angle of \(18^{\circ}\) with the horizontal. The new ramp will make an angle of \(12^{\circ}\) with the horizontal.

Step 2 - Determine the height of the ramp from the ground using the original angle

Using the sine function: \[ h = 10 \sin(18^{\circ}) = 10 \times 0.309 = 3.09 \text{ feet} \]

Step 3 - Calculate the length of the new ramp

The height remains the same, so use the sine function with the new angle: \[ \text{Length of new ramp} = \frac{h}{\sin(12^{\circ})} = \frac{3.09}{0.208} \approx 14.86 \text{ feet} \]

Key Concepts

Understanding the Sine Function

The sine function is one of the primary trigonometric functions and it plays an essential role in angle and length calculations in right-angled triangles.
In any right-angled triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
Mathematically, this is written as: \( \text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
For instance, in our problem, where we have a ramp, \( \text{sin}(18^{\theta}) = \frac{\text{height}}{10 \text{ feet (original ramp length)}} \).
It's important to use the sine function whenever you need to find the height or hypotenuse of a right triangle when one angle and one side length are known.

Steps for Accurate Angle Calculation

Calculating angles correctly is vital when working with trigonometry. Here's how we can determine the height of a ramp using an angle:
First, identify the given angle and the length of the hypotenuse (which is the length of the ramp).
Next, use the sine function to find the height. For the original ramp, this calculation is \( h = 10 \times \text{sin}(18^{\theta}) \).
Using a calculator, \( \text{sin}(18^{\theta}) \) is approximately 0.309.
So, multiply 10 (the ramp length) by 0.309 which gives 3.09 feet.
This height is crucial for calculating the length of the new ramp using the new angle.

Ramp Length Calculation Simplified

Now we need to find the length of the new ramp that makes a different angle with the horizontal.
First, keep in mind that the height of the ramp remains the same (3.09 feet).
We use the sine function in reverse to calculate the hypotenuse (the new ramp length).
We have: \( \text{Hypotenuse} = \frac{\text{Height}}{\text{sin}(\text{angle})} \)
So, for the new ramp, it would be: \( \text{Length of new ramp} = \frac{3.09}{\text{sin}(12^{\theta})} \)
Given that \( \text{sin}(12^{\theta}) \) is approximately 0.208, we then have: \( \frac{3.09}{0.208} \) which equals approximately 14.86 feet.
Therefore, the new ramp should be about 14.86 feet long.
This method ensures the ramp’s height aligns perfectly with the previous height, making the transition between angles smooth and accurate.