Question

GIVEN: Spider leaves the ground at a horizontal distance of 36 Inches from the observer. (Assume observer's eye is at ground level.) The spider climbs a straight vertical silk thread at 2 inches/minute.

FIND: Rate the angle of elevation (observer's view) increases when the spider is 36 inches above the ground by answering each of these questions:
a) SKETCH and label a right triangle using angle of elevation, and x height of spider.

Answer 1

To find the rate at which the angle of elevation increases, use the tangent function to find the relationship between the height of the spider and the horizontal distance. Then, differentiate the tangent function to find the rate of change.

Step-by-step explanation:

To find the rate at which the angle of elevation increases, let's sketch a right triangle using the angle of elevation and the height of the spider. The horizontal distance between the observer and the spider is given as 36 inches. Let's label the height of the spider as 'x'.

Since the spider is climbing a straight vertical silk thread, the right triangle formed by the observer, the spider, and the ground is a right triangle. The angle of elevation is the angle between the observer's line of sight and the ground. The angle opposite to the height of the spider is the angle of elevation. The rate at which the angle of elevation increases can be found by using the tangent function.

Tan(angle of elevation) = height of spider / horizontal distance

Tan(angle of elevation) = x / 36 inches

Now, we can find the rate at which the angle of elevation increases by finding the derivative of the tangent function with respect to time.