Exam 29: Applied Applications of the Derivative

Find the volume of the largest right cylinder that can be inscribed in a cone of height $20.00 \mathrm{~cm}$ . and a base radius of $15.00 \mathrm{~cm}$ .

The charge through a resistor is given by $q=2.93 t^{3}-3.98 t$ . Write an expression for the instantaneous current through the resistor. Evaluate at $t=4.00 \mathrm{~s}$ .

A spherical snowball is melting at $120 \mathrm{~cm}^{3} / \mathrm{min}$ . At what rate is the surface area decreasing at when the radius is $15 \mathrm{~cm}$ ?

Find the maximum area for an isosceles triangle with a perimeter of $76.2 \mathrm{~cm}$ .

Separate the number 200 into two parts such that the sum of one part times 60 and square of the second part is a minimum.

The end of a robotic arm follows the displacement equation $s=2 t^{3}-3 t^{2}+t-3$ where $\mathrm{s}$ is in $\mathrm{mm}$ . What is the acceleration at the end of the arm after $1.5 \mathrm{~s}$ ?

What dimensions of a rectangular beam will give the maximum strength? The beam is to be cut from a circular log with a diameter of $80.0 \mathrm{~cm}$ ? $\left(S=k x y^{2}\right)$

A pebble is dropped into a calm pond causing a series of circular ripples. The radius of the outer ripple is increasing at $0.700 \mathrm{~m} / \mathrm{s}$ . How fast is the area increasing at $5.00 \mathrm{~s}$ ?

The voltage applied to a $0.250-F$ capacitor is given by $v=15.0 t^{2}-48.0 t+40.0$ . Find the current at $6.25 \mathrm{~s}$ .

The angular displacement of a rotating object is given by $\theta=\frac{1}{4} t\left(t^{2}+1\right)^{3}$ in radians. Find the angular acceleration of the object at $2.400 \mathrm{~s}$ .

A submarine passes $30 \mathrm{~m}$ beneath a boat while traveling at $8.89 \mathrm{~m} / \mathrm{s}$ on a course of $\mathrm{N} 34^{\circ} 21^{\prime} \mathrm{W}$ . The boat continues on a heading of $\mathrm{N} 55^{\circ} 39^{\prime} \mathrm{E}$ at a speed of $13.3 \mathrm{~m} / \mathrm{s}$ . (a) How far do radio signals between the two craft have to travel 3 s later? (b) How fast are the two craft moving apart at this time?

A point has horizontal and vertical displacements (in $\mathrm{cm}$ ) of $x=3 t^{3}+5 t$ and $y=13+5 t^{2}$ . Find the $x$ and $y$ components of the velocity and acceleration at $t=2.00 \mathrm{~s}$ .

What value of $x$ will give the maximum efficiency of a screw with a coefficient of friction $\mu=0.82$ ? $E=\frac{x-\mu x^{2}}{x+\mu}$

A point is moving along the curve $y^{2}=3 x^{2}-12 x$ so that its $y$ -value increases at a constant rate of 3 units/s. How fast is the $x$ -value changing when the $x$ -value of the point increases to 6 ?

The voltage applied to a $0.25-\mathrm{F}$ capacitor is given by $v=\frac{\left(t^{2}-2 t\right)^{3}}{5 t}$ . Find the current at $3.7 \mathrm{~s}$ .

A point has horizontal and vertical displacements (in $\mathrm{cm}$ ) of $x=t^{3}-t^{2}+5 t$ and $y=24-t^{3}+t$ . Find the magnitude and direction of the resultant velocity at $t=3.50 \mathrm{~s}$ .

The displacement (in $\mathrm{cm}$ ) of a point is given by $s=4 t^{3}-3 \sqrt{1+2 t}$ . Find the instantaneous velocity and acceleration at $t=1.5 \mathrm{~s}$ .

A rope runs through a pulley $3.05 \mathrm{~m}$ off the ground. One end hangs straight down and is attached to a car's engine that is being hoisted. The other end is attached $0.61 \mathrm{~m}$ off the ground to the rear bumper of another car. The car drives away at $4 \mathrm{~km} / \mathrm{h}$ . How fast, in $\mathrm{m} / \mathrm{s}$ , is the rope going through the pulley when the car's rear bumper is $4.88 \mathrm{~m}$ from being under the pulley? The rope is long enough so that the engine is between the ground and the pulley.

The power $(\mathrm{W})$ delivered to a load by a $40 \mathrm{~V}$ source with internal resistance of $5 \Omega$ is $P=40 i-5 i^{2}$ . What current will give the maximum power?

A rectangular hardwood dance floor has one border against a carpeted area and the rest bordering tile. If the border along the carpeted edge costs twice as much as the border with the tiles, what are the dimensions of a $80.0 \mathrm{~m}^{2}$ dance floor with minimum border costs?