The spread of oil leaking from a tanker is in the shape of a circle. If the radius r (in feet) of the spread after t hours is $r(t)=200 \sqrt{t},$ find the area A of the oil slick as a function of the time t.

An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 22 meters per day. Express the area of the circle as a function of d, the number of days elapsed.

Consider a rectangular bathtub whose base is

$$18 ft^2.$$

At what rate is water pouring into the tub if the water level rises at a rate of 0.8 ft/min?

Consider a rectangular bathtub whose base is

$$18 ft^2.$$

How fast is the water level rising if water is filling the tub at a rate of

$$0.7 ft^3/min?$$

Determine the value of the literal numbers in each of the given matrix equalities. If the matrices cannot be equal, explain why.

$$\begin{bmatrix} x \\ x+y \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 4 &-3 \end{bmatrix}$$

Given that

$$\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$$

with convergence in (−1, 1), find the power series for the function with the given center a, and identify its interval of convergence. $f(x)=\frac{1}{1-2 x} ; a=0$

A refrigerator has a coefficient of performance of 2.25, runs on an input of 135 W of electrical power, and keeps its inside compartment at $5^{\circ} \mathrm{C}$. If you put a dozen 1.0-L plastic bottles of water at $31^{\circ} \mathrm{C}$ into this refrigerator, how long will it take for them to be cooled down to $5^{\circ} \mathrm{C}$? (Ignore any heat that leaves the plastic.)

Multiply or divide as indicated. The restrictions on the variables are implied.

$$\dfrac{3a^5b^7}{a-5b}\cdot\dfrac{2a-10b}{12a^4b^{10}}$$

I have 100 items of a product in stock. The probability mass function for the product’s demand D is $P(\mathrm{D}=90)=P(\mathrm{D}=100)=P(\mathrm{D}=110)=\frac{1}{3}$. a. Find the mass function, mean, and variance of the number of items sold. b. Find the mass function, mean, and variance of the amount of demand that will be unfilled because of lack of stock.

Express each repeating decimal as a fraction. (a)

$$5 . \overline { 23 }$$

(b)

$$1.3 \overline { 7 }$$

(c)

$$2.1 \overline { 35 }$$

For two invertible $n \times n$
matrices $A$ and $B$, determine the formulas stated below is necessarily true:

$A + B$ is invertible, and$(A + B)^{-1} = A^{-1} + B^{-1}$

Let $L$ be the set of all strings, including the null string, that can be constructed by repeated application of the following rules: If $\alpha\in L$, then $a\alpha b\in L$ and $b\alpha a\in L$. If $\alpha\in L$ and $\beta\in L$, then $\alpha\beta\in L$.

Prove that if $\alpha$ has equal number of $a$'s and $b$'s, then $\alpha\in L$.

Does the graph of $- 8 x ^ { 2 } - 12 = 3 - y$ open upward or downward?

Suppose that p(x) is the density function for heights of American men, in inches. What is the meaning of the statement p(68)=0.02?