Convert the rectangular equation to polar form. Assume $a>0$.

$$y^2=x^3$$

The pressure of a sample of helium in a 1.00-L container is 0.988 atm. What is the new pressure if the sample is placed in a 2.00-L container?

Find $dy/dx$ and $d^2y/dx^2$, and find the slope and concavity (if possible) at the given value of the parameter.

$$\underline{Parametric\ Equation}$$

$$x=\sqrt t,\ y=3t-1$$

$$\underline{Point}$$

$$t=1$$

Area Find the dimensions of the rectangle (with sides parallel to the coordinate axes) of maximum area that can be inscribed in the ellipse $x^2/25+y^2/16=1$.

Consider a projectile launched at a height h feet above the ground and at an angle θ with the horizontal. When the initial velocity is v_0 feet per second, the path of the projectile is modeled by the parametric equations x = (v_0 cos θ)t and y = h + (v_0 sin θ) t - 16t². -The center field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit 3 feet above the ground. It leaves the bat at an angle of θ degrees with the horizontal at a speed of 100 miles per hour. (a) Write a set of parametric equations for the path of the ball. (b) Use a graphing utility to graph the path of the ball when θ = 15°. Is the hit a home run? (c) Use a graphing utility to graph the path of the ball when θ = 23°. Is the hit a home run? (d) Find the minimum angle at which the ball must leave the bat in order for the hit to be a home run.

Graphical Reasoning Consider the parabola $x^2=4 p y$

Explain how the result of part (c) can be used as a sketching aid when graphing parabolas.

Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately $0.967$. The length of the major axis of the orbit is approximately $35.88$ astronomical units. (An astronomical unit is about $93$ million miles.)

Find the greatest (aphelion) and smallest (perihelion) distances from the sun’s center to the comet’s center.

Graphical Reasoning Consider the parabola $x^2=4 p y$

For each parabola in part (a), find the length of the latus recturn (see figure). How can the length of the latus rectum be determined directly from the standard form of the equation of the parabola?

Antireflection coatings for glass usually have an index of refraction that is less than that of glass. Explain how this permits a thinner coating.

Long-distance radio navigation for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light ($186,000$ miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola having the transmitting stations as foci. Assume that two stations, $300$ miles apart, are positioned on the rectangular coordinate system at points with coordinates $(-150,0)$ and $(150,0)$ and that a ship is traveling on a hyperbolic path with coordinates $(x,75)$

Determine the distance between the ship and station 1 when the ship reaches the shore.

Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately $0.967$. The length of the major axis of the orbit is approximately $35.88$ astronomical units. (An astronomical unit is about $93$ million miles.)

Use a graphing utility to graph the equation of the orbit.

Graphical Reasoning Consider the parabola $x^2=4 p y$

Locate the focus for each parabola in part (a).

Graphical Reasoning Consider the parabola $x^2=4py$.

Use a graphing utility to graph the parabola for $p=1,p=2,p=3,\ \text{and}, p=4$ and Describe the effect on the graph when increases.

Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately $0.967$. The length of the major axis of the orbit is approximately $35.88$ astronomical units. (An astronomical unit is about $93$ million miles.)

Find an equation of the orbit. Place the center of the orbit at the origin, and place the major axis on the $x$-axis.