Question

2 - 24. a trundle wheel (a wheel attached to the end of stick) is used to measure distance. as you walk with the wheel on the ground, it makes a click after each revolution. a. if the wheel has a radius of one foot, what is the exact distance for one revolution? b. what would the radius need to be so that each revolution is equal to exactly 10 feet? 2 - 25. without using a calculator, determine if each of the given functions is increasing or decreasing on the interval x = 1 to x = 2. verify your answer by sketching the graph. a. $f(x)=x^{2}$ b. $g(x)=-2x + 1$ c. $h(x)=1 - x^{3}$

Explanation:

2 - 24

a.

Step1: Recall circumference formula

The distance for one - revolution of a circle is its circumference, and the formula for the circumference of a circle is $C = 2\pi r$.

Step2: Substitute radius value

Given $r = 1$ foot, substituting into the formula $C=2\pi(1)=2\pi$ feet.

Step1: Set up the equation

We know that the circumference $C = 2\pi r$, and we want $C = 10$ feet. So, the equation is $10=2\pi r$.

Step2: Solve for $r$

Divide both sides of the equation by $2\pi$: $r=\frac{10}{2\pi}=\frac{5}{\pi}$ feet.

Step1: Calculate function values

When $x = 1$, $f(1)=1^{2}=1$. When $x = 2$, $f(2)=2^{2}=4$. Since $f(2)>f(1)$, the function is increasing on the interval $x = 1$ to $x = 2$.

Step2: Analyze the graph

The graph of $y = x^{2}$ is a parabola opening upwards with vertex at $(0,0)$. On the interval $(1,2)$, as $x$ increases, $y$ also increases.

Answer:

$2\pi$ feet

b.