Question

Danielle is a civil engineer for Dastis Dynamic Construction, Inc. She must create blueprints for a wheelchair accessible ramp leading up to the entrance of a mall that she and her group are building. The ramp must be exactly 100 meters in length and make a \(20^{\circ}\) angle with the level ground. What is the horizontal distance, in meters, from the start of the ramp to the point level with the start of the ramp immediately below the entrance of the mall, rounded to the nearest meter? (Note: Disregard units when inputting your answer, \(\sin 20^{\circ} \approx 0.324, \cos 20^{\circ} \approx\) \(\left.0.939, \tan 20^{\circ} \approx 0.364\right)\)

Short answer

The horizontal distance from the start of the ramp to the point level with the start of the ramp immediately below the entrance of the mall is approximately 94 meters.

Step-by-step solution

Identify the known values and the unknown value

In this problem, we are given the total length of the ramp (hypotenuse) to be 100 meters and the angle it makes with the ground to be 20 degrees. We are to find the horizontal distance from the start of the ramp to the point level with the start of the ramp immediately below the entrance of the mall (the adjacent side of the right-angled triangle formed). We will use trigonometric functions to determine the unknown value.

Find the trigonometric function to use

Since we know the angle and the hypotenuse, we need to find the adjacent side of the right-angled triangle. We can use the cosine function to find the adjacent side. The cosine function is defined as: \(cos(\theta) = \frac{adjacent}{hypotenuse}\)

Calculate the adjacent side (horizontal distance)

We can now substitute the known values and solve for the adjacent side: \(cos(20^{\circ}) = \frac{adjacent}{100}\) We are also given the approximate value for \(cos(20^{\circ})\), which is 0.939. Now, we can calculate the adjacent side: \(0.939 = \frac{adjacent}{100}\) Multiplying both sides by 100, we get: \(adjacent = 0.939 \times 100\)

Round the answer to the nearest meter

Calculating the adjacent side, we get: \(adjacent \approx 93.9\) Now we round the answer to the nearest meter: Adjacent side (horizontal distance) ≈ 94 meters So, the horizontal distance from the start of the ramp to the point level with the start of the ramp immediately below the entrance of the mall is approximately 94 meters.

Key concepts

Trigonometry SAT Questions

Understanding trigonometry is key to excelling in SAT math, specifically when tackling trigonometric SAT questions. These questions assess your ability to interpret and apply trigonometric ratios such as sine, cosine, and tangent. In the context of the provided exercise, which involves finding the horizontal distance of a ramp, cosine function comes into play since you have the hypotenuse and need to find the adjacent side.

To handle trigonometry-based SAT questions, remember the basic definitions: for a right-angled triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side. Using these definitions, you can solve for missing sides or angles of a right-angled triangle, as demonstrated in the solution provided.

Also, it's crucial to remember the approximate values of the trig functions at common angles, as provided in the exercise, as well as how to round your answers appropriately, to ensure accuracy and efficiency during the test.

SAT Math Practice

Regular SAT math practice is essential to hone your problem-solving skills and speed. One effective method of practice is to work through problems that require the use of trigonometric functions, as real SAT questions will often test your understanding of these concepts, just like in our example about the wheelchair ramp.

When practicing:

• Focus on understanding the relationships between the sides and angles of triangles.
• Memorize the values of trigonometric functions for commonly used angles.
• Practice problems that require converting between different trigonometric functions.
• Use real-life applications, like the ramp scenario, to visualize and understand the concepts better.

Moreover, use the official College Board materials or other reputable resources for practice questions that mimic the format and difficulty level of SAT questions. This targeted practice can greatly improve your confidence and performance on the actual test.

SAT Problem Solving

SAT problem solving, particularly in math, requires a clear understanding of concepts, the ability to apply knowledge to new situations, and critical thinking skills. When faced with a trigonometry problem, there is a step-by-step approach to find the solution:

Step 1: Analyze the Question

Understand what is being asked. Identify the known and unknown quantities, visualize the problem (e.g., by drawing a diagram), and decide which trigonometric ratio to use.

Step 2: Apply the Appropriate Trigonometric Function

Based on what you need to find (adjacent side, opposite side, or hypotenuse) and what you know (given angle, any side of the triangle), apply the relevant trigonometric function.

Step 3: Solve for the Unknown

Substitute the known values into the trigonometric equation and solve for the unknown quantity.

Step 4: Verify and Round

Check your answer and ensure it makes sense in the context of the problem. Then round your answer if required.

Remember, practice is paramount to develop a quick and accurate problem-solving approach. When solving SAT math problems, always consider the most efficient method to get to the solution, manage your time effectively, and double-check your work for careless errors.