Question

Ryan and Allison build a ramp to help their elderly cat, Simms, walk up to their bed. They need the ramp to make a \(35^{\circ}\) angle with their bedroom floor. How long must the ramp be to reach the top of their bed that is exactly three feet off the ground? A) \(\frac{\sin 35^{\circ}}{3}\) B) \(\frac{\sin 55^{\circ}}{3}\) C) \(\frac{3}{\sin 55^{\circ}}\) D) \(\frac{3}{\sin 35^{\circ}}\)

Short answer

D) \(\frac{3}{\sin 35^{\circ}}\)

Step-by-step solution

Identifying the given information

We have a right triangle with one angle at \(35^\circ\) and the height of the triangle (opposite side) is 3 feet. We need to find the length of the ramp (hypotenuse).

Write the sine formula

The sine of an angle in a right triangle is equal to the length of the opposite side divided by the length of the hypotenuse. We can write this as: \[\sin(\angle) = \frac{\text{opposite}}{\text{hypotenuse}}\]

Plug in the given information

We know the angle and the length of the opposite side, so we can plug them into the sine formula: \[\sin(35^\circ) = \frac{3}{\text{hypotenuse}}\]

Solve for hypotenuse

To find the length of the hypotenuse, we can rearrange the formula and solve for it: \[\text{hypotenuse} = \frac{3}{\sin(35^\circ)}\]

Find the correct answer

Comparing the expression for the hypotenuse to the given options, we see that the answer is: D) \(\frac{3}{\sin 35^{\circ}}\)

Key concepts

Right Triangle

A right triangle is one of the most fundamental shapes in geometry, characterized by having one angle exactly equal to 90 degrees. The sides of a right triangle have unique names: the longest side opposite the right angle is called the hypotenuse, and the other two sides are referred to as the legs, with one being the adjacent leg or base (next to the angle in question), and the other being the opposite leg (across from the angle in question).

Understanding right triangles is crucial for solving trigonometry problems because they allow the use of trigonometric ratios—sine, cosine, and tangent—which relate the angles of a right triangle to the lengths of its sides. For example, in the given problem, Ryan and Allison are using a right triangle formed by the ramp, the height of the bed, and the floor to determine the length of the ramp based on the required angle and height.

In problems like these, visualizing or sketching the triangle can be immensely helpful for identifying which sides and angles are involved in the calculations and applying the correct trigonometric ratios.

Sine Formula

The sine formula is a powerful tool when solving right triangle problems. It's based on the sine function, which in a right triangle, is the ratio of the length of the opposite side to the length of the hypotenuse. The formula can be expressed as: \[\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\]

In our exercise, the opposite side is the height from the floor to the top of the bed, and we're looking for the hypotenuse, which will be the length of the ramp. When given the angle and the length of the opposite side, you can rearrange the sine formula to solve for the hypotenuse: \[\text{hypotenuse} = \frac{\text{opposite}}{\sin(\theta)}\]

This rearrangement allows us to isolate the hypotenuse on one side of the equation, making it easier to solve for. By understanding and correctly applying the sine formula, students can answer questions like the one in our exercise, where Ryan and Allison need to calculate the length of a ramp for their cat, Simms.

SAT Mathematics Preparation

Preparing for the mathematics section of the SAT involves familiarizing yourself with a wide range of topics, including algebra, geometry, and notably, trigonometry. Problems involving right triangles and the sine formula, like the one in our exercise, are common. To perform well, it's necessary to master not just the computational skills but also the problem-solving strategies.

• Understand the concepts: Ensure you have a strong grasp of trigonometric ratios and how they apply to right triangles.
• Practice problems: Regularly solve practice problems to become comfortable with different types of questions and formats.
• Memorize key formulas: Knowing formulas like the sine, cosine, and tangent ratios off the top of your head can save valuable time during the test.
• Time management: Learn to manage your time effectively by quickly identifying the approach for each problem and avoiding spending too long on any single question.
• Review mistakes: Go over practice tests to understand where you went wrong and how to correct your approach.

By dedicating time to mastering these areas, especially trigonometry problems involving right triangles and the sine formula, test-takers can significantly boost their SAT math scores and increase their chances of a higher overall SAT result.