Experiments show that in a temperature field, heat flows in the direction of maximum decrease of temperature T. Find this direction in general and at the given point P. Sketch that direction at P as an arrow. T=x²+y²+4z², P:(2, -1, 2)

Regular sunscreen use is an important part of keeping your skin healthy by protecting it from ultraviolet radiation, but some chemical agents in sunscreens may be harmful if absorbed into the skin. Research is ongoing to find less harmful alternatives and to understand the factors involved in skin absorption. The IUPAC names are given below for two common chemical sunscreens. Provide a structure for each:

1-(4-Methoxyphenyl)-3-[4-(1,1-dimethylethyl)phenyl]propane-1,3-dione (also known as avobenzone).

Use a calculator to find each trigonometric ratio to three significant digits.

cot 23.3 deg

In a recent study of the hourly wages of maintenance crew members for major airlines, the mean hourly wage was found to be $20.50, with a standard deviation of$3.50. Assuming the distribution of hourly wages follows the normal probability distribution, if we select a crew member at random, what is the probability the crew member earns between $20.50 and$24.00 per hour?

Write the ratio as a fraction in simplest form. 28. 18 girls to 27 boys

Find the direction cosines and angles of u, and demonstrate that the sum of the squares of the direction cosines is 1. u = ⟨0, 6, -4⟩

Write the ratio as a fraction in simplest form. 15. 16 dogs to 12 cats

Sketch the graphs of the given functions by determining the appropriate information and points from the first and second derivatives. Use a calculator to check the graph.

$$y = x^3 - 12x + 12$$

Write the linear function $f$ such that it has the indicated function values. Then sketch the graph of the function.

$$f(0)=-5, f(4)=-8$$

The integrative functions of the nervous system are performed mainly by a. afferent neurons. b. efferent neurons. c. neuroglia. d. sensory neurons. e. interneurons.

Determine each estimate without using a calculator. Then use a calculator to perform the computation necessary to obtain an exact answer. How reasonable is your estimate when compared to the actual answer? If a person who works 40 hours per week earns $61,500 per year, estimate that person's hourly wage.

Find a unit vector that has:

(a) The same direction as $\textbf{a}$; and

(b) A direction opposite to that of $\textbf{a}$.

$$\textbf{a}=\left<1,2,2\right>$$

For the following problem, find the direction cosines and direction angles for the position vector to the given point.

(5,7,-1)

Two rocks collide with each other in outer space, far from all other objects. Rock 1 with mass 5 kg has velocity $\langle 30,45,-20\rangle\ \mathrm{m} / \mathrm{s}$ before the collision and $\langle- 10,50,-5\rangle\ \mathrm{m} / \mathrm{s}$ after the collision. Rock 2 with mass 8 kg has velocity $\langle- 9,5,4\rangle\ \mathrm{m} / \mathrm{s}$ before the collision. Calculate the final velocity of rock 2.