Math Problem Statement

A tent pole, of length 226 cm, stands perpendicular to the ground. A straight support rod connects the top of the tent pole to a point on the ground. The angle between the support rod and the ground is 63°. Calculate the length of the support rod. Give your answer in centimetres to the nearest integer.

Solution

To solve the problem, we can use trigonometry since the scenario forms a right triangle.


Step 1: Understand the problem

  1. Given:
    • Length of the tent pole (opposite side): 226cm226 \, \text{cm}
    • Angle between the support rod and the ground: 6363^\circ
  2. We are tasked to calculate the length of the support rod (the hypotenuse).

Step 2: Identify the trigonometric relationship

Since we know the angle and the opposite side and want the hypotenuse, we use the sine function:

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

Here:

  • θ=63\theta = 63^\circ
  • Opposite = 226cm226 \, \text{cm}
  • Hypotenuse = LL (the length of the support rod).

Rearranging for LL:

L=oppositesin(θ)L = \frac{\text{opposite}}{\sin(\theta)}


Step 3: Substitute values

Substitute θ=63\theta = 63^\circ and opposite = 226cm226 \, \text{cm}:

L=226sin(63)L = \frac{226}{\sin(63^\circ)}

Using a calculator to find sin(63)\sin(63^\circ):

sin(63)0.8910\sin(63^\circ) \approx 0.8910

Now calculate LL:

L=2260.8910253.7L = \frac{226}{0.8910} \approx 253.7


Step 4: Round the answer

The length of the support rod, rounded to the nearest integer, is:

254cm\boxed{254 \, \text{cm}}


Final Answer:

The support rod is 254 cm long.


Let me know if you would like me to explain this further or solve a similar problem! Here are 5 related questions:

  1. How would the problem change if the angle were 4545^\circ instead of 6363^\circ?
  2. How can we calculate the distance of the base of the support rod from the pole?
  3. What trigonometric function relates the adjacent side and the hypotenuse?
  4. If the tent pole were taller, say 300cm300 \, \text{cm}, how would the support rod length change?
  5. How do you use the cosine rule to find unknown side lengths in triangles?

Tip:

Always check which trigonometric function (sine, cosine, tangent) applies to a triangle problem based on the sides and angles you know.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangles
Sine Function

Formulas

sin(θ) = opposite / hypotenuse
L = opposite / sin(θ)

Theorems

Basic Trigonometric Ratios

Suitable Grade Level

Grades 9-10