Farmer Abe has a budget of $300 to build a rectangular fence to protect his rambunctious sheep. He decides that three sides of the fence will be constructed with chain-link, which costs only $1 per foot. However, the fourth side of the fence faces his neighbor, whom Farmer Abe hates with a passion. To make his neighbor angry, Farmer Abe decides that the fourth side of the fence will be constructed as a giant mirror facing outward, which costs $5 per foot. Find the dimensions of the largest area the fence can enclose.
A window has the shape of a rectangle with two semicircles attached, giving it an oval shape as seen in the picture below. If the window is to have a perimeter of 28 ft, what should its dimensions be in order to allow the maximum amount of light through it (that is, the maximum area)? Describe the resulting shape.
A cylindrical tank is being filled simultaneously with water and sugar. The input of water and sugar is carefully adjusted so that the concentration of sugar in the water is held constant at 5 grams per cubic meter. The tank has a radius of 5 meters and a height of 10 meters. At a particular point in time, the water level is 3 meters high and rising at a rate of 0.5 meters per second. At this time, how quickly is the mass of sugar in the tank increasing?
Sketch the graph of a function f having the following properties: Label points of inflection, if any, on the graph.
The domain of f(x) is (-∞, 5) ∪ (5, ∞) and f is continuous on its domain.
f(-6) = -3; f(2) = 4.
x = 5 is a vertical asymptote.
lim f(x) = -∞ and lim f(x) = 0.
x→-∞ x→∞
f'(2) = 0; f'(x) > 0 on (-∞, 2) and on (5, ∞); f'(x) < 0 on (2, 5).
f''(x) < 0 on (-∞, 3) and on (5, ∞); f''(x) > 0 on (3, 5).