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How does pressure change in a combustion engine? How fast does the water level rise when filling a pool? Calculus quantifies the impact of change on areas, angles, distances, temperatures, and more.

Word Problems - Basic

A \(13\text{ feet}\) long ladder is leaning against a wall and sliding toward the floor. If the foot of the ladder is sliding away from the base of the wall at a rate of \(17\text{ feet/sec},\) how fast is the top of the ladder sliding down the wall (in feet/sec) when the top of the ladder is \(5\text{ feet}\) from the ground?

At time \(t=0 \mbox{ s}\), the radius of a circle is equal to \(19 \mbox{ cm}\). The radius of the circle increases at a rate of \(\frac{1}{2} \ \mbox{cm/s}\). The rate of change of area at \(t=18 \mbox{ s}\) is equal to \(m\pi \ \mbox{cm}^2\mbox{/s} \), where \(m\) is a positive integer. What is the value of \(m\)?

Throwing a stone in a calm lake forms concentric ripples, as shown in the above picture. If the radius of one ripple increases at the rate of \(10 \text{cm}\) per second, what is the rate of increase of the area of the ripple \(2\) seconds after a stone is thrown (in \(\text{cm}^2/\text{s}\))?

During a rainstorm, the area of a circular pool of water increases at the rate of \(4000\pi\text{ in}^2\text{/sec}.\) At what rate is the radius expanding (in inches/sec) when the radius is \(200\text{ inches}?\)

A point is moving along the parabola \(x+5y^2=0\) toward the origin with \(x\)-coordinate increasing at the rate of \(5\text{ units/s}\) and \(y\)-coordinate positive. How fast is the point moving in terms of the remaining distance to the origin as it passes through the point \((-5,1)\) on the parabola?


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