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In Exercises 7-26, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation if necessary.
$x=\textrm{ln}\ 2t$$y=2t^{2}$
In Exercises 33-40, use the results of Exercises 29-32 to find a set of parametric equations for the line or conic.
Ellipse: vertices: $(3, 7), (3, -1); \quad$ foci: $(3, 5), (3, 1)$
PROJECTILE MOTION A projectile is launched at a height of $h$ feet above the ground at an angle of $\theta$ with the horizontal. The initial velocity is $v_0$ feet per second, and the path of the projectile is modeled by the parametric equations
$x= (v_0 \cos\ \theta)t \quad \textrm{and} \quad y=h+(v_0 \sin\ \theta)t - 16t^2.$
In Exercises 61 and 62, use a graphing utility to graph the paths of a projectile launched from ground level at each value of $\theta$ and $v_0$. For each case, use the graph to approximate the maximum height and the range of the projectile.
(a) $\theta = 60^{\circ}, \quad v_0 = 88$ feet per second(b) $\theta = 60^{\circ}, \quad v_0 = 132$ feet per second(c) $\theta = 45^{\circ}, \quad v_0 = 88$ feet per second(d) $\theta = 45^{\circ}, \quad v_0 = 132$ feet per second
(a) $\theta = 15^{\circ}, \quad v_0 = 50$ feet per second(b) $\theta = 15^{\circ}, \quad v_0 = 120$ feet per second(c) $\theta = 10^{\circ}, \quad v_0 = 50$ feet per second(d) $\theta = 10^{\circ}, \quad v_0 = 120$ feet per second
SPORTS The center field fence in Yankee Stadium is 7 feet high and 408 feet from home plate. A baseball is hit at a point 3 feet above the ground. It leaves the bat at an angle of $\theta$ degrees with the horizontal at a speed of 100 miles per hour (see figure).
(a) Write a set of parametric equations that model the path of the baseball.(b) Use a graphing utility to graph the path of the baseball when $\theta = 15^{\circ}$. Is the hit a home run?(c) Use the graphing utility to graph the path of the baseball when $\theta = 23^{\circ}$. Is the hit a home run?(d) Find the minimum angle required for the hit to be a home run.
The graph of a function $f$ is given. Estimate $\int_{0}^{8} f(x) d x$ using four subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints.
Janine is 19 years old. She opens an account that pays 4.4% interest compounded annually. She sets a goal of saving $2,000 each year. By the time she is 30 years old, how much will her account be worth? Include your work with your answer.
Mary wants to go on a $10,000 vacation in 6 months. She has a bank account that pays 4.25% interest, compounded monthly. How much must she deposit each month to afford the vacation?
What happens to the internal pressure in a fluid flowing in a horizontal pipe when its speed increases?
If a journal entry includes a debit or credit to the Retained Earnings account, it is most likely which of the following?
Which of the following managerial accounting techniquesattempts to allocate manufacturing overhead in a more meaningfulfashion?
A group of 125 truck owners were asked what brand of truck they owned and whether or not the truck has four-wheel drive. The results are summarized in the two-way table below.
for while drive yes no totalford 28 17 45cheay 32 18 50dodge 20 10 30total 80 45 125
Randomly select one truck owner. What is the probability that he or she owns a Ford or has four-wheel drive? P(Ford ? 4WD)?
