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THIRD HOUR EXAM, MATH 223
Instructor: Kris Green, April 10, 1997

1.

Multiple Choice (5 points each):

(a)

The determinant of the Jacobian for the change of coordinates x = -u², y = u-v+w, z = u-v-w is

a. 0 b. 4

c. -4u d. 4 u

(b)

Given the line parameterized by x = 2t - 1, y = 3 - (5/3)t, z=2, a vector pointing in the direction of the line is

$$2\hat{i} - (5/3) \hat{j} + 2\hat{k} -2\hat{i} + (5/3) \hat{j} - 2\hat{k}$$

$$2\hat{i} - (5/3) \hat{j} -\hat{i} + 3 \hat{j} + 2\hat{k}$$

(c)

The plane passing through the point (-1,3,2) containing the vectors $2\hat{i} - 5\hat{j}$ and $\hat{i} + 3\hat{j} - 2\hat{k}$ is parameterized by:

a. x = -1 + 2p + q, y=3-5p + 3q, z = 2-2q

b. x = -1 + 3p, y = 3 + 2p, z = 2- 2p

c. x = 1 - p +2q, y = 3 + 3p - 5q, z = -2 + 2p

d. x = -1 + p +2q, y = 3 + 3p - 5q, z = 2 - 2q

(d)

The curve $\vec{r}(t) = t^2 \hat{i} + (t^2 - 4)\hat{j} + 2t\hat{k}$intersects the sphere x² + y² + z² = 16 at time:

a. t = 0 b. t = 1

$$(2,-2,\pm2\sqrt{2})$$ d. both (a) and (c) e. both (a) and (b)

2.

Fox television, for no apparent reason, has asked vector calculus students to compute the following double integral. First, sketch the region of integration. Next, switch the order of integration, and then compute the new integtral. Give your answer exactly.

$$\int_1^2 \int_0^y \sin (x^2) dx dy$$

3.

The density of fish in a lake is given by the density function $\delta (x,y,z) = -z(z + 16)$ (z = 0 is the surface of the lake.) The bottom of the lake is in the shape of the function z = x² + y² - 16 and lies in the range $-16 \le z \le 0$. To find the total population of fish in the lake, first sketch the region of integration (the lake,) set up the triple integral in cylindrical coordinates, and compute the integral to find the total number of fish. (Hint: there is an easy way to do the integral over r using a u substitution.)

4.

A long time ago, in a galaxy far, far away, the planet Qix orbited the star Caleb II. The position of Qix with respect to the star is given by the parametric function

$$\vec{r}_Q (t) = 3 \cos (2\pi t/320) \hat{i} + (4 + 5\sin(2 \pi t/320)) \hat{j}$$

where t is measured in days. The moon of Qix orbits the planet in a circle at a distance 0.5 andcompletes an orbit once every eight days.

(a)

Sketch the orbit of Qix.

(b)

As seen from the planet, write the parametric equations for the motion of the moon around Qix.

(c)

Now, place the origin of your coordinates at Caleb II and use vector addition to help you write the parametric equations for the motion of the moon around the star.

(d)

EXTRA CREDIT (5 points):

i.

How many days does it take Qix to complete an orbit? (ie. how long is a year on Qix?)

ii.

How many times does the moon orbit Qix in one year?

5.

A pyrotechnician named Beavis launches a Flaming Dragon firework into the air and, after it has reached its maximum height, finds that its position is given parametrically by:

$$\vec{r} (t) = 12 \cos (t) \hat{i} + 12 \sin (t) \hat{j} + (1000-5t)\hat{k}$$

(a)

Find the speed of the Flaming Dragon as a function of time.

(b)

Find the acceleration of the Flaming Dragon as a function of time.

(c)

How far does the Flaming Dragon travel along its path from the time it reaches its maximum height (labelled t = 0) until it explodes 10 seconds later?

(d)

Into what shape and color does the Flaming Dragon explode?