Math Problem Statement

Trey wants to use string lights to build a flagpole lighted tree in his front yard. Trey already has a 7-foot-tall pole mounted in the ground. If the angle each light string makes with the ground is 73.5°73.5°, use the sine ratio to determine how long each light string must be to reach the top of the pole. Round the answer to the nearest tenth.(1 point)

Solution

To solve this problem using the sine ratio, we need to relate the height of the pole and the angle with the length of the string. The sine ratio is given by:

sin(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

Here:

  • θ=73.5\theta = 73.5^\circ (the angle the string makes with the ground),
  • Opposite = height of the pole = 7 feet,
  • Hypotenuse = length of the string = xx (to be determined).

Rewriting the equation:

sin(73.5)=7x\sin(73.5^\circ) = \frac{7}{x}

Solve for xx:

x=7sin(73.5)x = \frac{7}{\sin(73.5^\circ)}

Using a calculator, sin(73.5)0.95882\sin(73.5^\circ) \approx 0.95882:

x=70.958827.3feet.x = \frac{7}{0.95882} \approx 7.3 \, \text{feet}.

Final Answer:

The length of each light string must be approximately 7.3 feet.


Would you like more details or have any questions?
Here are 5 related questions for further practice:

  1. What would the string length be if the angle was 60° instead of 73.5°?
  2. How does the length change if the height of the pole is doubled?
  3. Can cosine or tangent ratios be used for this problem? Why or why not?
  4. What is the horizontal distance of the string from the base of the pole?
  5. If Trey uses 5 strings evenly spaced around the pole, how many feet of lights will he need in total?

Tip: Always double-check trigonometric calculations to ensure accuracy, especially when rounding intermediate values.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Sine Ratio
Right Triangles

Formulas

sin(θ) = opposite / hypotenuse

Theorems

Sine ratio in right triangles

Suitable Grade Level

Grades 8-10