Understanding the Rate of Change of Angle Theta

The video tutorial explains how to find the rate of change of the angle theta formed by a wire attached to a 60-foot pole. It begins by setting up the problem and identifying the right triangle formed. The sine function is used to express theta in terms of x, the wire length. The tutorial then differentiates this expression using the chain rule to find d(theta)/dx. The derivative is simplified by clearing fractions under the square root. Finally, the derivative is evaluated for a 100-foot wire, yielding a result of -0.0075 radians per foot.

What is the height of the pole in the problem?

• 80 feet

• 60 feet

• 70 feet

• 50 feet

In the context of the problem, what does the wire represent in the right triangle?

• Base

• Adjacent side

• Opposite side

• Hypotenuse

How is theta expressed as a function of x?

• Theta = sin(x/60)

• Theta = tan(60/x)

• Theta = inverse sine(60/x)

• Theta = cos(60/x)

Which mathematical rule is applied to differentiate theta with respect to x?

• Power rule

• Chain rule

• Quotient rule

• Product rule

What is the derivative of 60 times x to the power of negative one?

• 60/x

• 60x

• -60/x^2

• -60x^2

Why is it beneficial to clear the fraction from under the square root in the derivative?

• To make the expression longer

• To add more variables

• To simplify the expression

• To make it more complex

What is the simplified form of the derivative function?

• 60/(x * sqrt(x^2 + 3600))

• 60/(x^2 - 3600)

• -60/(x^2 + 3600)

• -60/(x * sqrt(x^2 - 3600))

What is the rate of change of theta when the wire length is 100 feet?

• -0.0075 radians per foot

• 0.0075 radians per foot

• -0.075 radians per foot

• 0.075 radians per foot

How is the rate of change of theta expressed in decimal form?

• -0.0075

• 0.0075

• -0.075

• 0.075

What is the fractional representation of the rate of change of theta?

• 3/40

• -3/40

• -3/400

• 3/400