Question

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Explanation:

Problem 24

To find the sine ratio of \(\angle Z\) in the right triangle \(XYZ\) (right - angled at \(Y\)), we use the definition of the sine function in a right triangle. The sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

• For \(\angle Z\), the opposite side to \(\angle Z\) is \(XY\) with length \(16\), and the hypotenuse is \(XZ\) with length \(20\).
• By the definition of sine, \(\sin(\angle Z)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{XY}{XZ}=\frac{16}{20}\)

Step 1: Identify the triangle type and formula

The wheelchair ramp, the vertical rise, and the horizontal extension form a right - triangle. We can use the Pythagorean theorem, which states that for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c = \sqrt{a^{2}+b^{2}}\). Here, \(a = 2\) feet (vertical rise) and \(b = 10\) feet (horizontal extension), and \(c\) is the length of the ramp.

Step 2: Substitute values into the formula

Substitute \(a = 2\) and \(b = 10\) into the Pythagorean theorem:
\(c=\sqrt{2^{2}+10^{2}}=\sqrt{4 + 100}=\sqrt{104}\)

Step 3: Calculate the value and round

\(\sqrt{104}\approx10.2\) (rounded to the nearest tenth)

Answer:

A. \(\frac{16}{20}\)

Problem 25