A lens has one convex surface of radius $6.00 \mathrm{~cm}$ and one concave surface of radius $10.0 \mathrm{~cm}$. When an object is placed $35.0 \mathrm{~cm}$ from the lens, a real image is formed $50 \mathrm{~cm}$ from the lens.
(a) What is the focal length of the lens?
(b) What is the index of refraction of the lens?

If a verb in the following sentences is incorrect, cross it out and write the correct form above it. If the verb is already correct, write C above it.

Example 1. The chef $\overset{\textit{\color{#c34632}{set}}}{\sout{\text{sat}}}$ an assortment of ingredients on the counter.

She had set on a high stool at the work counter.

a. Solve the system by linear combination:\

$$\ \begin{aligned} &5 x-8 y=47 \\ &2 x+7 y=-22 \end{aligned}$$

b. Solve the system by determinants:\

$$\begin{array}{lr} 4 x-3 y= & 5 \\ 7 x+2 y= & -1 \end{array}$$

A solar cooker consists of a curved reflecting surface that concentrates sunlight onto the object to be warmed . The solar power per unit area reaching the Earth's surface at the location is $600 \mathrm{~W} / \mathrm{m}^2$. The cooker faces the Sun and has a diameter of $0.600 \mathrm{~m}$. Assume that $40.0 \%$ of the incident energy is transferred to $0.500 \mathrm{~L}$ of water in an open container, initially at $20.0^{\circ} \mathrm{C}$. How long does it take to completely boil away the water? (Ignore the heat capacity of the container.)

Use properties of determinants to evaluate the given determinant by inspection. Explain your reasoning.

$$\left| \begin{array} { r r r } { 0 } & { 0 } & { 1 } \\ { 0 } & { 5 } & { 2 } \\ { 3 } & { - 1 } & { 4 } \end{array} \right|$$

Some populations initially grow exponentially but eventually level off. Equations of the form $P(t)=\frac{M}{1+A e^{-k t}}$ where M, A, and k are positive constants, are called logistic equations and are often used to model such populations. Here M is called the carrying capacity and represents the maximum population size that can be supported, and $A=\frac{M-P_{0}}{P_{0}}$, where $P_0$ is the initial population. Compute $\lim _{M \rightarrow \infty} P(t)$. What kind of function is your result?

The demand function for a product is modeled by p=20-0.02x, $0 \leq x \leq 1000$ where p is the price (in dollars) and x is the number of units. (a) Determine when the demand is elastic, inelastic, and of unit elasticity. (b) Use the result of part (a) to describe the behavior of the revenue function.

At the instant shown, the 11,000-kg airplane's velocity is $\mathbf{v}=3001(\mathrm{~m} / \mathrm{s})$. The rate of change of the magnitude of the velocity is $d v / d t=5 \mathrm{~m} / \mathrm{s}^2$. The radius of curvature of the airplane's path is $4500 \mathrm{~m}$, and the $y$ axis points toward the concave side of the path. The thrust is $T=120,000 \mathrm{~N}$. Determine the lift $L$ and drag $D$.

Use the table along with dimensional analysis, to convert the given unit to the unit indicated. Where necessary, round to two decimal places.

$$\begin{matrix} \text{Volume in Cubic Units} & \text{Capacity}\\ \text{1 }{\mathrm{cm}^{3}} & \text{ = 1 mL}\\ \text{1 }{\mathrm{dm}^{3}=1000 \mathrm{cm}^{3}} & \text{ = 1 L}\\ \text{1 }{\mathrm{m}^{3}} & \text{ = 1 kL}\\ \end{matrix}$$

9 pt to L

A laser beam is incident at an angle of $30.0 ^ { \circ }$ to the vertical onto a solution of corn syrup in water. If the beam is refracted to $19.24 ^ { \circ }$ to the vertical, Find speed in the solution.

Newton’s law of gravity says that the gravitational force between two bodies is attractive and given by

$$F=\frac{GMm}{r^{2}},$$

where $G$ is the gravitational constant, $m$ and $M$ are the masses of the two bodies, and $r$ is the distance between them. This is the famous inverse square law. For a falling body, we take $M$ to be the mass of the earth and $r$ to be the distance from the body to the center of the earth. So, actually, $r$ changes as the body falls, but for anything we can easily observe (say, a ball dropped from the Tower of Pisa), it won’t change significantly over the course of the motion. Hence, as an approximation, it is reasonable to assume that the force is constant. According to Newton’s second law, acceleration is caused by a force and

$$\text{Force}=\text{Mass}\ \times\ \text{Acceleration}.$$

(b) Explain how the differential equation shows the acceleration of the body is independent of its mass.

Compute the determinants using cofactor expansion along the first row and along the first column.

$$\left| \begin{array} { l l l } { 1 } & { 1 } & { 0 } \\ { 1 } & { 0 } & { 1 } \\ { 0 } & { 1 } & { 1 } \end{array} \right|$$

Use properties of determinants to evaluate the given determinant by inspection. Explain your reasoning.

$$\left| \begin{array} { l l l } { 1 } & { 2 } & { 3 } \\ { 0 } & { 4 } & { 1 } \\ { 1 } & { 6 } & { 4 } \end{array} \right|$$

We use the vertical line test to determine whether a graph is of a function. What do you know about the definition of a function, explain the vertical line test.