Work each problem. Find a value of c so that $y=x^2-10x+c$ has exactly one x-intercept.

Perform each operation. Write answers in standard form. $(4-3 i)^{2}$

In a study designed to investigate the effects of a strong magnetic field on the early development of mice, ten cages, each containing three 30-day-old albino female mice, were subjected for a period of 12 days to a magnetic field having an average strength of 80 Oe/cm. Thirty other mice, housed in ten similar cages, were not put in the magnetic field and served as controls. Listed in the table are the weight gains, in grams, for each of the twenty sets of mice.

$$\scriptstyle \begin{array}{c} \hline \text { In Magnetic Field } & \text { Not in Magnetic Field } \\ \hline\end{array}$$

$$\scriptstyle\begin{array}{cccc} \text { Cage } & \text { Weight Gain (g) } & \text { Cage } & \text { Weight Gain (g) } \\ \hline 1 & 22.8 & 11 & 23.5 \\ 2 & 10.2 & 12 & 31.0 \\ 3 & 20.8 & 13 & 19.5 \\ 4 & 27.0 & 14 & 26.2 \\ 5 & 19.2 & 15 & 26.5 \\ 6 & 9.0 & 16 & 25.2 \\ 7 & 14.2 & 17 & 24.5 \\ 8 & 19.8 & 18 & 23.8 \\ 9 & 14.5 & 19 & 27.8 \\ 10 & 14.8 & 20 & 22.0 \\ \hline \end{array}$$

Test whether the variances of the two sets of weight gains are significantly different. Let $\alpha=0.05 .$ For the mice in the magnetic field, $s_{X}=5.67 ;$ for the other mice, $s_{Y}=3.18$.

Particle accelerators, such a s the Large Hadron Collider, use magnetic fields to steer charged particles a round a ring. Consider a proton ring with 36 identical bending mag nets connected by straight segments. The protons move along a 1.0-m-long circular arc as they pass through each magnet. What magnetic field strength is needed i n each magnet to steer protons around the ring with a speed of $2.5 \times 10^{7} \mathrm{m} / \mathrm{s}$? Assume that the field is uniform inside the magnet, zero outside.

Find the relative rate of change of f(x) at the indicated value of x. Round to three decimal places. f(x)=420-5 x; x=25

Suppose an experimenter intends to do a regression analysis by taking a total of 2n data points, where the $x_{i}$'s are restricted to the interval [0,5]. If the xy-relationship is assumed to be linear and if the objective is to estimate the slope with the greatest possible precision, what values should be assigned to the $x_{i}$'s?

Monochromatic light at 577 nm illuminates a diffraction grating with 325 lines/mm. Determine the highest order that can be observed with this grating at the given wavelength.

Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. $7(5 t+3)^{-5 / 3}+14(5 t+3)^{-2 / 3}-21(5 t+3)^{1 / 3}$

Work each problem. Let U={1, 2, 3, 4, 5} and A={1, 2, 3}. Find A'.

Find $f^{\prime}(x)$. $f(x)=3 x^{e}-2 e^{x}$

Refer to the n=5 data points $\left(x_{1}, y_{1}\right)=(0,4)$ $\left(x_{2}, y_{2}\right)=(1,5),\left(x_{3}, y_{3}\right)=(2,7),\left(x_{4}, y_{4}\right)=(3,9)$, and $\left(x_{5}, y_{5}\right)=(4,13)$. Calculate the indicated sum or product of sums. $\sum_{k=1}^{5} x_{k} y_{k}$

Seven different financing plans with their D-E mixes and costs of debt and equity capital for a new innovations project are summarized below. Use the data to determine what mix of debt and equity capital will result in the lowest WACC. $$ \begin{array} \text{ } & \text{Debt Capital} & \text{ } & \text{Equity Capital} & \text{ }\\ \hline \text{Plan} & \text{Percentage} & \text{Rate, \\%} & \text{Percentage} & \text{Rate, \\%}\\ \hline \text{1} & \text{100} & \text{14.5} & \text{-} & \text{-}\\ \text{2} & \text{70} & \text{13.0} & \text{30} & \text{7.8}\\ \text{3} & \text{65} & \text{12.0} & \text{35} & \text{7.8}\\ \text{4} & \text{50} & \text{11.5} & \text{50} & \text{7.9}\\ \text{5} & \text{35} & \text{9.9} & \text{65} & \text{9.8}\\ \text{6} & \text{20} & \text{12.4} & \text{80} & \text{12.5}\\ \text{7} & \text{-} & \text{-} & \text{100} & \text{12.5}\\ \hline \end{array} $$

A large crate of mass m is placed on the back of a truck but not tied down. As the truck accelerates forward with an acceleration a, the crate remains at rest relative to the truck. What force causes the crate to accelerate forward? the force of gravity.

You are given the 2012 value of a product and the rate at which the value is expected to change during the next 5 years. Use this information to write a linear equation that gives the dollar value V of the product in terms of the year. (Let t=12 represent 2012.)

$$\begin{matrix} \text{2012 Value} & \text{Rate}\\ \text{\$245,000} & \text{\$5600 increase per year}\\ \end{matrix}$$