Problem 1: Maximize the area of a rectangle inside any right triangle with base=B and height=H.
Find the dimensions w and h of the rectangle and the area of the rectangle with maximum area that can fit inside the right triangle (as shown in the diagram). Clearly label each result.
Tip: Find a relationship (equation) between w and h and then solve that equation for one or the other variable. Then, substitute that expression into the expression for area to get the formula for the area in terms of only one of the variables.
Show your expression for area (in terms of one variable) to a Lab Leader.
How does the area of the rectangle compare to the area of the triangle?
Check your solution for the problem graphically, by plotting the expression for the area of the rectangle vs w when H=20 and B=15. You should be able to see a clear maximum area.
What is the value of w at that point (as compared to B)?
Problem 2: Volume of revolution problem.
The Duriset Paving Company needs to purchase a new cement mixer. The company needs a cement mixer large enough to hold cement for an entire driveway. Jennifer, the CEO of this Fortune 100 paving company, has done research to determine that the average driveway is 50 feet long, 15 feet wide, and one-quarter of a foot deep. Using these dimensions, you can help Jennifer determine the proper size of the mixer. The shape of the mixer is given by the curve:
Plot r(z) vs z for z from 0 to 10 to see what the curve looks like. Use the scaling=constrained option in your plot command to make the z- and r-axes have the same units.
Rotating the function r(z) around the z-axis will allow you to calculate the volume of the mixer. To visualize the mixer with a 3D-plot of the mixer, enter the following commands into Maple (where rz is the name of the expression your created earlier representing r(z) ). This is a parametric 3D plot.
plot3d([ z, rz*cos(t), rz*sin(t) ], z=0..10, t=0..2*Pi, scaling=constrained)with(plots):or here's code to animate the figure:
animate(plot3d, [[z,rz*cos(t),rz*sin(t)], z=0..10, t=0..T, scaling=constrained], T=0..2*Pi)To run the animation: click somewhere in the plot, then locate and try out the animation control buttons that appear at the top of the worksheet.
What is the volume of an average driveway as described above?
Set up and enter the integral that computes the volume of a mixer of length, L.
Determine the length of the mixer needed to hold enough cement for an average driveway. Tip: set your integral expression equal to your average volume and solve for the length of the mixer.
Jennifer discovers that mixers only come in lengths of 10, 15, and 20 feet. Compute the excess volume for each mixer that is longer than needed.
Which mixer(s) will hold enough cement for the average driveway? Show this in Maple and to a Lab Leader.
Problem 3: Gabriel's Horn
Gabriel's Horn is an interesting surface that challenges our intuition about volume and surface area. Gabriel's Horn is the solid of revolution obtained by revolving the area of the region bounded by y=1/x and the x-axis for x from 1 to ∞ about the x-axis.
Recall that the equations for the surface area and volume of a function rotated about the independent variable (x-axis in most cases) are:
Plot Gabriel's Horn. Hint: start with the parametric equations from Problem 2 and modify them accordingly.
Compute the surface area of Gabriel's Horn.
Compute the volume of Gabriel's Horn.
Taking the volume and surface area into account, what is interesting about Gabriel's Horn? If Gabriel's Horn were used to hold paint, how much paint would it hold? How much paint would it take to cover the surface?
Show your results to a Lab Leader.
Cylinder in a cone
The figure from Problem 1 can also be looked at as the cross-section of a cylinder inside a cone.
Redo Problem 1, but now consider it as a cylinder of radius w and height h fitting in a cone. In this case B is the radius of the base of the cone and H is the height. Find the cylinder of maximum volume that will fit inside the cone. The formula for the volume of a cylinder of radius r and height h is πr2h . The volume of a cone with base radius r and height h is (1/3) πr2h.
What is the ratio of the volume of the cylinder to the volume of the cone? How does this compare with the rectangle in a triangle? Does this seem reasonable?