ASSESSMENT 42 WEEK 4 Solving Measures of the | StudyX - Free Printable

Here are the step-by-step solutions for each question on the worksheet.

Questions 1 & 2

*Refer to the figure with the kite.*

1. What is the trigonometric ratio to be used to find the measure of the unknown side?
* Step 1: Identify the sides relative to the given angle ($70^\circ$).
* The side labeled $x$ is opposite the $70^\circ$ angle.
* The side labeled $65\text{ m}$ is the hypotenuse (the longest side, opposite the right angle).
* Step 2: Choose the correct ratio using SOH CAH TOA.
* We have the Opposite side ($x$) and the Hypotenuse ($65$).
* The ratio that uses Opposite and Hypotenuse is Sine ($\sin$).
* Formula: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
* Conclusion: This matches option A.

2. How far is the kite from the ground?
* Step 1: Set up the equation using the sine ratio identified in Question 1.
$$ \sin(70^\circ) = \frac{x}{65} $$
* Step 2: Solve for $x$. Multiply both sides by 65.
$$ x = 65 \times \sin(70^\circ) $$
* Step 3: Calculate the value.
$$ \sin(70^\circ) \approx 0.9397 $$
$$ x \approx 65 \times 0.9397 \approx 61.08 $$
* Conclusion: The height is $61.08\text{ m}$. This matches option C.

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Questions 3 - 5

*Refer to the figure with the building.*

3. What is the unknown side in the right triangle?
* Step 1: Look at the position of side $x$ relative to the given angle ($30^\circ$).
* Step 2: Side $x$ is directly across from the $30^\circ$ angle. In a right triangle, the side across from the angle is called the "opposite" side.
* Conclusion: The unknown side is the opposite side. This matches option C.

4. What is the correct equation in solving the unknown side?
* Step 1: Identify the knowns and unknowns.
* Angle: $30^\circ$
* Adjacent side (next to the angle): $50\text{ m}$
* Opposite side (unknown): $x$
* Step 2: Choose the ratio. We have Opposite and Adjacent, so we use Tangent ($\tan$).
$$ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} $$
* Step 3: Plug in the numbers.
$$ \tan(30^\circ) = \frac{x}{50} $$
* Conclusion: This matches option B.

5. What is the height of the building to the nearest whole number?
* Step 1: Use the equation from Question 4: $\tan(30^\circ) = \frac{x}{50}$.
* Step 2: Solve for $x$.
$$ x = 50 \times \tan(30^\circ) $$
* Step 3: Calculate the value.
$$ \tan(30^\circ) \approx 0.5774 $$
$$ x \approx 50 \times 0.5774 \approx 28.87 $$
* Step 4: Round to the nearest whole number.
$28.87$ rounds up to $29$.
* Conclusion: The height is $29\text{ m}$. This matches option D.

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Question 6

6. Approximately, what is the measure of $\angle A$ in the figure below?
* Step 1: Identify the sides relative to angle $A$.
* Opposite side: $8\text{ ft}$
* Adjacent side: $10\text{ ft}$
* Step 2: Choose the ratio. We have Opposite and Adjacent, so we use Tangent ($\tan$).
$$ \tan(A) = \frac{8}{10} = 0.8 $$
* Step 3: Find the angle using the inverse tangent function ($\tan^{-1}$).
$$ A = \tan^{-1}(0.8) $$
* Step 4: Calculate the angle.
$$ A \approx 38.6598^\circ $$
* Step 5: Round to the nearest hundredths.
$$ 38.66^\circ $$
* Conclusion: This matches option A.

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Final Answer:
1. A
2. C
3. C
4. B
5. D
6. A