Suman Saurav T.

GEOMETRY A line segment through a focus of an ellipse with endpoints on the ellipse and perpendicular to the major axis is called a latus rectum of the ellipse. Therefore, an ellipse has two latera recta. Knowing the length of the latera recta is helpful in sketching an ellipse because it yields other points on the curve (see figure). Show that the length of each latus rectum is $2b^2/a$.

Find the average rate of change of the function over the given interval or intervals.
$$
P(\theta)=\theta^{3}-4 \theta^{2}+5 \theta ; \quad[1,2]
$$

Inflating a balloon The volume $V=(4 / 3) \pi r^{3}$ of a spherical balloon changes with the radius.
a. At what rate $\left(\mathrm{ft}^{3} / \mathrm{ft}\right)$ does the volume change with respect to the radius when $r=2 \mathrm{ft} ?$
b. By approximately how much does the volume increase when the radius changes from 2 to 2.2 $\mathrm{ft}$ ?

You are planning to make an open rectangular box from an 8 -in.-by-15-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of
the box of largest volume you can make this way, and what is its volume?

Consider a 100 -N weight suspended by two wires as shown in the accompanying figure. Find the magnitudes and components of the force vectors $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ .

Sailboat The wind passing over a boat's sail exerted a 1000 -lb
magnitude force $\mathbf{F}$ as shown here. How much work did the wind
perform in moving the boat forward 1 mi? Answer in foot-pounds.