SPRING 2016 MATH 152 LAB ASSIGNMENT #5
DUE: MAY 2, 2016
1. Chapter #8
#11. A 20 ft-long rod is cut into 12 pieces, which are welded together to form the frame of a rectangular box. The length of the
box’s base is 15 in. longer than its width.
(a) Create a polynomial expression for the volume V in terms of x.
(b) Make a plot of V versus x.
(c) Determine the x that maximizes the volume and determine that
volume.
#12. A rectangular piece of cardboard, 40 in. long by 22 in. wide, is
used for making a rectangular box (open top) by cutting out squares
of x by x from the corners and folding up the sides.
(a) Create a polynomial expression for the volume V in terms of x.
(b) Make a plot of V versus x.
(c) Determine x if the volume of the box is 1, 000 in.3 .
(d) Determine the value of x that corresponds to the box with the
largest possible volume, and determine that volume.
#23. Growth data of a sunflower plant is given in the following table:
(a) Curve-fit the data with a third-order polynomial. Use the polynomial to estimate the
height in week 6.
(b) Fit the data with linear and spline interpolations and use each interpolation to estimate the height in week 6.
1
2
DUE: MAY 2, 2016
In each part make a plot of the data points (circle markers) and the fitted curve or the
interpolated curves. Note that part (b) has two interpolation curves.
#25. The standard air density, D (average of measurements made), at different heights,
h, from sea level up to a height of 33 km is given below.
(a) Make the following four plots of the data points (density as a function of height): (1)
both axes with linear scale; (2) h with log axis, D with linear axis; (3) h with linear axis, D with log axis; (4) both log axes. According to the plots choose a function
(linear, power, exponential, or logarithmic) that best fits the data points and determine the coefficients of the function.
(b) Plot the function and the points using linear axes.
#26. Write a user-defined function that fits data points to a power function of the form
y = bxm . Name the function [b, m] = powerf it(x, y), where the input arguments x and
y are vectors with the coordinates of the data points, and the output arguments b and m
are the constants of the fitted exponential equation. Use powerfit to fit the data below.
Make a plot that shows the data points and the function.
2. Chapter #9
#4. Determine the positive roots of the equation x2 − 5x sin(3x) + 3 = 0.
SPRING 2016 MATH 152 LAB ASSIGNMENT #5
3
#12. Determine the dimensions (radius r and height h) and the volume of the cylinder with the largest volume that can be made inside
of a sphere with a radius R of 14 in..
#27. The orbit of Pluto is elliptical in shape, with a = 5.9065 ×
109 km and b = 5.7208 × 109 km. The perimeter of an ellipse can be
calculated by
Z π/2 p
P = 4a
1 − k 2 sin2 θdθ
0
a2 −b2
a .
where k =
Determine the distance Pluto travels in one orbit. Calculate the average speed at which Pluto travels (in km/h) if one orbit takes about 248 years.
#32. Use a MATLAB built-in function to numerically solve:
dy
x3 e−y
= −x2 +
for 1 ≤ x ≤ 5 with y(1) = 1
dx
4
Plot the solution.
#39. Tumor growth can be modeled with the equation
ν dA
A
= αA 1 −
dt
k
where A(t) is the area of the tumor and α, k, and ν are constants. Solve the equation for
0 ≤ t ≤ 30 days, given α = 0.8, k = 60, ν = 0.25, and A(0) = 1 mm2 . Make a plot of A
as a function of time.