Question

to find the distance across a lake, a surveyor took the measurements in the figure shown. use these measurements to determine how far it is across the lake. a = \square yd (round the answer to the nearest whole number.)

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle \( \triangle ABC \) with \( \angle C = 90^\circ \), \( AC = 690 \) yd, and \( \angle A = 47^\circ \). We need to find the length of \( BC = a \). We can use the tangent function, which is defined as \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). Here, \( \tan(47^\circ)=\frac{a}{690} \).

Step2: Solve for \( a \)

Rearrange the formula to solve for \( a \): \( a = 690\times\tan(47^\circ) \). Calculate \( \tan(47^\circ) \approx 1.0724 \). Then \( a = 690\times1.0724 \approx 739.956 \).

Step3: Round to the nearest whole number

Rounding \( 739.956 \) to the nearest whole number gives \( 740 \).

Answer:

740