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Math 230 - Homework 5

Be sure to show all your work and clearly indicate your final answer. Word problems should be answered in complete sentences. Please follow these markers:

[C] - You should use a computer to either graph (e.g. on geogebra) or do calculations I can’t expect you to do by hand (e.g. on wolframAlpha). Do not use a calculator on problems without this symbol.

[P] - A picture is required to receive full credit for this problem; the picture should be labeled with relevant information.

[S] - Your answer needs to be given in the form of a complete sentence, with units where applicable.

1. Find all solutions for the following systems of equations. State your answers as We will be solving systems of equations like this in class soon, so if you are unsure how to do this, please ask in office hours soon.

2. [P] Find and sketch the domain of the following functions.

3. [P] Consider the function which has the contour map below.

Draw a sketch of the graph from the side. Sketch a 3d version of the graph as well if you can.

4. [S] The function W(h, A) represents the weight (in kg) of a bull elephant as a function of its height (h, in meters) and age (A, in years).

(a) Give a real-life interpretation for the expression W(2.5, 38) = 4,500, including proper units.

(b) Give a real-life interpretation for the expression including proper units.

(c) Give a real-life interpretation for the equation W(h, 45) = 4,800; what would a solution to this equation tell us?

5. [C] Consider a function for the height (in cm) of a plant after 1 month, based on the following collected data. (Across the bottom are mL of water, and down the left side are mg of fertilizer.)

(a) If we gave giving 20 mL of water and 4 mg of fertilizer, then approximate the rate at which the height of the plant would change if we kept the amount of fertilizer the same, but increased the amount of water. (Give your answer with appropriate units.)

(b) If we gave giving 20 mL of water and 4 mg of fertilizer, then approximate the rate at which the height of the plant would change if we kept the amount of water the same, but increased the amount of fertilizer. (Give your answer with appropriate units.)

(c) Approximate the height a plant would reach after 1 month if we gave 22 mL of water and 4.5 mg of fertilizer.

6. [P] In , we are throwing a ball at t = 0 from the point (0, h) towards the positive x direction. Our initial speed will be 10 m/s and our initial velocity vector makes an angle of θ with the x-axis. The only acceleration acting on the ball is due to gravity, which we can approximate as Our goal is to make a formula which will tell us where the ball lands, as a function of h and θ.

(a) What is the initial velocity of the ball?

(b) Find a parametric function for the position of the ball at time t; leave h and θ as unknown constants throughout.

(c) Find the time T at which the ball hits the x-axis as a function of h and θ.

(d) Find the x-coordinate where the ball hits the x-axis as a function of h and θ.

(e) [S] [C] Calculate where the ball lands if launched from a height of 5.3 meters and with an angle of 37◦.

7. [S] Let be a function where a population of cells, P, depends on the ambient temperature, T, in degrees Celsius, and the availability of a liquid “food”, F, in mL.