Exam 29: Applied Applications of the Derivative

A car drives towards a $111 \mathrm{~m}$ skyscraper at $7.62 \mathrm{~m} / \mathrm{s}$ . How fast is the car approaching the top of the building when the car is $68.6 \mathrm{~m}$ from the base of the building?

The distance $(\mathrm{m})$ that a point travels in $t$ seconds is given by $s=\sqrt{5+t^{2}}$ . Find the velocity and acceleration of the point at $2 \mathrm{~s}$ .

A metal bar $4.00 \mathrm{~cm}$ by $6.00 \mathrm{~cm}$ by $48.0 \mathrm{~cm}$ is being heated such that each length is increasing at 0.05 $\mathrm{cm} / \mathrm{s}$ . At what rate is the volume increasing?

Split the number 15 into two parts so that the product of one part and the other is maximized.

A light is located $20.0 \mathrm{ft}$ above the ground. A person $6.00 \mathrm{ft}$ tall walks away from the light at a rate of $3.00 \mathrm{ft} / \mathrm{s}$ . Find the rate at which the person's shadow is increasing when the person is $10.0 \mathrm{ft}$ from the spot directly under the light.

A plane moves according to $s=225 t^{2}-2.5 t^{3}$ , where $t$ is in minutes and $s$ in metres. What is the maximum speed attained by the plane?

The displacement in metres of an arrow shot straight up is giving by $s=35 t-4.9 t^{2}$ , where $t$ is in seconds. Find the maximum height reached by the arrow.

A circular plate is cooled and contracts so its radius is shrinking by $0.01 \mathrm{~cm} / \mathrm{s}$ . How fast is the area of one face decreasing when the radius is $2.5 \mathrm{~cm}$ ?

What is the maximum volume of cylindrical tin can that can be made from $750.0 \mathrm{~cm}^{2}$ of metal?

The temperature $T\left({ }^{\circ} \mathrm{C}\right)$ at distance $x(\mathrm{~mm})$ from the end of a certain heated bar is given by $T(x)=2.0 x^{1.5}$ $+4.0 x+7.0$ . Find the rate of change of temperature with respect to distance at the point $55 \mathrm{~mm}$ from the end.

A point moves along the curve that has the parametric equations $x=8 t^{3}-3 t^{2}$ and $y=6 t^{3}+25, x$ and $y$ being in metres and $t$ being in minutes. Find the magnitude and direction of the acceleration when $t=2.50$ minutes.

The angular displacement in radians is given by $\theta=8 \sqrt{t^{2}+2}$ . Find the angular velocity and angular acceleration at $0.25 \mathrm{~s}$ .

If a voltage of $v=1.7 t^{2}-3.2 t+1.3$ is applied to a $250-\mu \mathrm{F}$ capacitor, what is the current when $0.3 \mathrm{sec}$ ?

A rocket fired at $275 \mathrm{~m} / \mathrm{s}$ at $36^{\circ}$ angle of elevation is described by $x=\left(275 \cos 36^{\circ}\right) t$ and $y=\left(275 \sin 36^{\circ}\right) t-4.9 t^{2}$ . Find the velocity (magnitude and direction) at $8 \mathrm{~s}$ .

What is the velocity at 3 seconds of an object with displacement $s=t^{3}-7 t \mathrm{~m}$ ?

A rectangular area with a certain size is to be enclosed on three sides by a fence and on one side by an existing building. Find the ratio of the sides of the area that minimizes the fence needed.

A point moves along the curve $y=2 x^{4}-10 x^{2}-x+2 \mathrm{~cm}$ . (a) Find the direction of travel at $x=2.01 \mathrm{~cm}$ . (b) If the speed of the point along the curve is $4.81 \mathrm{~cm} / \mathrm{s}$ , find the $x$ and $y$ components of the velocity when $x=2.01 \mathrm{~cm}$ .

Find the instantaneous velocity and acceleration at the given time for the straight-line motion described by the equation $s=12 t^{2}-t+15$ at $t=2.0 \mathrm{~s}$ , where $s$ is in centimetres.

Find the voltage across a $12.5-\Omega$ resistor at $2.50 \mathrm{~s}$ if the charge through the resistor is given by $q=8 \sqrt{4 t-1}$ .

A ship is travelling at $12.0 \mathrm{~km} / \mathrm{h}, 2.00 \mathrm{~km}$ parallel to shore. The ship passes a light house, what is the straight-line distance between the ship and the light house 30 minutes later?