A solar oven is to be made from an open box with reflective sides. Each box is made from a 30-in. by 24-in. rectangular sheet of aluminium with square of length x (in inches) removed from each corner. Then the flaps are folder up to form an open box. a. Show that the volume of the box is given by V x = 4 x 3 − 108 x 2 + 720 x for 0 < x < 12. b. Graph the function from part (a) and a “maximumâ€� feature on a graphing utility to approximate the length of the sides of the squares that should be removed to maximize the volume. Round to the nearest tenth of an inch. c. Approximate the maximum volume. Round to the nearest cubic inch.
A solar oven is to be made from an open box with reflective sides. Each box is made from a 30-in. by 24-in. rectangular sheet of aluminium with square of length x (in inches) removed from each corner. Then the flaps are folder up to form an open box. a. Show that the volume of the box is given by V x = 4 x 3 − 108 x 2 + 720 x for 0 < x < 12. b. Graph the function from part (a) and a “maximumâ€� feature on a graphing utility to approximate the length of the sides of the squares that should be removed to maximize the volume. Round to the nearest tenth of an inch. c. Approximate the maximum volume. Round to the nearest cubic inch.
Solution Summary: The author illustrates how a solar oven has to be made from an open box with reflective sides. The volume of the box is given by cvolume=lengthtimes width
A solar oven is to be made from an open box with reflective sides. Each box is made from a 30-in. by 24-in. rectangular sheet of aluminium with square of length x (in inches) removed from each corner. Then the flaps are folder up to form an open box.
a. Show that the volume of the box is given by
V
x
=
4
x
3
−
108
x
2
+
720
x
for
0
<
x
<
12.
b. Graph the function from part (a) and a “maximum� feature on a graphing utility to approximate the length of the sides of the squares that should be removed to maximize the volume. Round to the nearest tenth of an inch.
c. Approximate the maximum volume. Round to the nearest cubic inch.
2. Consider the following:
Prove that x, x2, and 1/x are the solutions to the homogeneous equation
corresponding to x³y"" + x²y" + 2xy' + 2y = 2x4.
b. use variation of parameters to find a particular solution and complete the general
solution to the differential equation. I am interested in process. You may use a
computer for integration, finding determinants and doing Kramer's.
3. A spring is stretched 6 in. by a mass that weighs 8 lb. The mass is attached to a dashpot
mechanism that has a damping constant of 0.25 lb-sec./ft. and is acted on by an external
force of 4 cos 2t lb.
a. Set-up the differential equation and initial value problem for the system.
b. Write the function in phase-amplitude form.
C.
Determine the transient solution to the system. Show your work.
d. Determine the steady state of this system. Show your work.
e.
Is the system underdamped, overdamped or critically damped? Explain what this
means for the system.
4. Suppose that you have a circuit with a resistance of 20, inductance of 14 H and a
capacitance of 11 F. An EMF with equation of E(t) = 6 cos 4t supplies a continuous charge
60
to the circuit. Suppose that the q(0)= 8 V and the q'(0)=7. Use this information to answer the
following questions
a. Find the function that models the charge of this circuit.
b. Is the circuit underdamped, overdamped or critically damped?
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