Answer :
Final answer:
To find the ground's angle of elevation (theta), we use trigonometry with the pole height, shadow length, and sun's angle of elevation. By calculating the inverse sine of the sun's angle, we find theta's value which is then rounded to match one of the choices provided.
Explanation:
To determine the angle of elevation of the ground (theta), we need to use trigonometric principles that involve the utility pole, its shadow, and the angle of elevation of the sun. When the sun's angle of elevation is 42 degrees, a 13-meter utility pole casts a 19-meter shadow down the slope. We can model this using right triangle trigonometry, with the utility pole as the perpendicular (opposite side to the angle of elevation), the shadow as the hypotenuse, and the angle between the hypotenuse and the ground as the angle of elevation we want to find. First, we calculate the angle theta using the inverse trigonometric functions. Knowing the length of the opposite side (pole height) and the hypotenuse (length of the shadow), we can use the sine function. The sine of the sun's angle of elevation (42 degrees) is equal to the opposite side (pole height) divided by the hypotenuse (shadow length):sin(42°) = opposite / hypotenuse = 13 / 19The inverse sine function (arcsin) can be used to find theta. Once calculated, you can round the result to one decimal place to find the best match among the options provided (A through D). Without providing explicit calculations here, the result should be one of the given options after finding theta and considering the ground's slope.
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