SPRING 2016 MATH 152 LAB ASSIGNMENT #1
DUE: FEB., 03
1. Chapter #1
5. Calculate:
7
◦
(a) cos( 7π
9 ) + tan( 15 π) sin(15 )
(b) sin2 80◦ −
(cos 14◦ sin 80◦ )2
√
3
0.18
17. In the triangle shown a = 5 in., b = 7 in., and γ = 25◦ . Define a,
b, and γ as variables, and then:
(a) Calculate the length of c by substituting the variables in the Law
of Cosines.
Law of Cosines: c2 = a2 + b2 − 2ab cos γ
(b) Calculate the angles α and β (in degrees) using the Law of Sines.
Law of Sines: sina α = sinb β = sinc γ
(c) Verify the Law of Tangents by substituting the results from part
(b) into the right and left sides of the equation.
Law of Tangents:
a−b
a+b
=
tan[ 21 (α−β)]
tan[ 12 (α+β)]
25. The voltage difference Vab between points a and b in the Wheatstone bridge circuit is:
R1 R3 − R2 R4
Vab = V
(R1 + R2 )(R3 + R4 )
Calculate the voltage difference when V = 14 volts, R1 = 120.6 ohms, R2 = 119.3 ohms,
R3 = 121.2 ohms, and R4 = 118.8 ohms.
38. The spread of a computer virus through a computer network can be modeled by:
N (t) = 20e0.15t
1
2
DUE: FEB., 03
where N (t) is the number of computers infected and t time in minutes.
(a) Determine how long it takes for the number of infected computers to double.
(b) Determine how long it takes for 1, 000, 000 computers to be infected.
40. Stirling’s approximation for large factorials is given by:
√
n
n! = 2πn( )n
e
Use the formula for calculating 20!. Compare the result with the true value obtained
with MATLAB’s built-in function f actorial by calculating the error (Error = (T rueV al−
ApproxV al)/T rueV al).
2. Chapter #11
4. Define x as a symbolic variable.
(a) Derive the equation of the polynomial that has the roots x = −2, x = −0.5, x = 2,
and x = 4.5.
(b) Determine the roots of the polynomial
f (x) = x6 − 6.5x5 − 58x4 + 167.5x3 + 728x2 − 890x − 1400
by using the f actor command.
8. A water tower has the geometry shown in the figure (the lower part
is a cylinder with radius R and height h, and the upper part is a half
sphere with radius R). Determine the radius R if h = 10m and the
volume is 1, 050m3 . (Write an equation for the volume in terms of the
radius and the height. Solve the equation for the radius, and use the
double command to obtain a numerical value.
13. The mechanical power output P in a contracting muscle is given
by:
v
kvT0 1 − vmax
P = Tv =
v
k + vmax
where T is the muscle tension, v is the shortening velocity (max of vmax ), T0 is the isometric tension (i.e., tension at zero velocity), and k is a non-dimensional constant that
ranges between 0.15 and 0.25 for most muscles. The equation can be written in nondimensional form:
ku(1 − u)
p=
k+u
where p = (T v)/(T0 vmax ), and u = v/vmax . Consider the case k = 0.25.
SPRING 2016 MATH 152 LAB ASSIGNMENT #1
3
(a) Plot p versus u for 0 ≤ u ≤ 1.
(b) Use differentiation to find the value of u where p is maximum.
(c) Find the maximum value of p.
16. Evaluate the following indefinite integrals:
R x3
dx.
(a) I = √1−x
R 2 2
(b) I = x cos x dx.
17. Define x as a symbolic variable and create the symbolic expression
cos2 x
1 + sin2 x
Rπ
Plot S in the domain 0 ≤ x ≤ π and calculate the integral I = 0
S=
cos2 x
1+sin2 x
dx.