Calculus Homework #8 Due 4-20

1. A spherical snowball is melting. Its radius is decreasing at 0.2 cm per hour when the radius is 15 cm. How fast is its volume decreasing at that time?

2. A  potter forms a piece of clay into a cylinder. As he rolls it, the length, L, of the cylinder increases and the radius, r, decreases. If the length of the cylinder is increasing at 0.1 cm per second, find the rate at which the radius is changing when the radius is 1 cm and the length is 5 cm.

3. A certain quantity of gas occupies a volume of 20cc at a pressure of 1 atmosphere. The gas expands without the addition of heat, so, for some constant k, its pressure, P, and volume, V, satisfy the equation PV^{1.4}=k. (a) Find the rate of change of pressure with respect to volume. (b) The volume is increasing at 2cc/min when the volume is 30cc. At that moment, is the pressure increasing or decreasing? How fast?

4. (a) A hemispherical bowl of radius 10cm contains water to a depth of hcm. Find the radius of the surface of the water as a function of h. (b) The water level drops at a rate of 0.1 cm per hour. At the what rate is the radius of the water decreasing when the depth is 5 cm?

Trigonometry Classwork 4-7

  1. Two planes leave an airport at the same time, one flying at 300 km/h and the other at 420 km/h. The angle between their flight paths is 75 degrees. After three hours, how far apart are they?
  2. From the top of a building 10 m tall, the angle of elevation to the top of a flagpole is 11 degrees. At the base of the building, the angle of elevation to the top of the flagpole is 39 degrees. Find the height of the flagpole.
  3. In quadrilateral ABCD, AB=3, BC=4, CD=5, and DA=6. The length of diagonal BD is 7. Find the length of the other diagonal.

Trigonometry Test 2 is on Tuesday!

There are 14 questions, broken down as follows.

3 graphing rational functions

2 simplifying logs

3 log equations

2 exponential equations

1 inverse function

1 graphing exponential function

1 graphing log function

1 write the equation of an exponential function given certain conditions

Here are some good review questions.

  1. Solve for x: \log_5(2x)=\log_5(x+1)-1
  2. Solve for x and express your solution as a ratio of natural logs: 3^{\frac{-x}{2}}=4^{x+1}
  3. Sketch the graph of f(x)=-\ln(x-4)+2
  4. Sketch the graph of f(x)=e^{\frac{-x}{3}+1}
  5. Sketch the graph of f(x)=\frac{(x+2)^2(x+1)}{(x-4)^3(x-6)}
  6. f(x) has a horizontal asymptote at y=5. Where is the asymptote for f^{-1}(x)?
  7. f(x)=-5^x+1. Write the equation for f^{-1}(x).
  8. Solve for x and express your solution as a ratio of natural logs: 3^x+2=10.
  9. Re-write as an exponential equation: \log_{64}4=\frac{1}{3}.
  10. Sketch the graph of f(x)=\frac{6x^2+19x-7}{2x+7}

Also study quizzes 2, 3, and 4.