Constructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets 12 inches wide. After marking off a length of 4 inches from each edge, the builder bends this length up at an angle θ . See the illustration. The area A of the opening as a function of θ is given by A ( θ ) = 16sin θ ( cos θ + 1 ) , 0 ∘ < θ < 90 ∘ (a) In calculus, you will be asked to find the angle that maximizes A by solving the equation cos ( 2 θ ) + cos θ = 0, 0 ∘ < θ < 90 ∘ Solve this equation for θ . (b) What is the maximum area A of the opening? (c) Graph A = A ( θ ) , 0 ∘ ≤ θ ≤ 90 ∘ and find the angle θ that maximizes the area A . Also find the maximum area.
Constructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets 12 inches wide. After marking off a length of 4 inches from each edge, the builder bends this length up at an angle θ . See the illustration. The area A of the opening as a function of θ is given by A ( θ ) = 16sin θ ( cos θ + 1 ) , 0 ∘ < θ < 90 ∘ (a) In calculus, you will be asked to find the angle that maximizes A by solving the equation cos ( 2 θ ) + cos θ = 0, 0 ∘ < θ < 90 ∘ Solve this equation for θ . (b) What is the maximum area A of the opening? (c) Graph A = A ( θ ) , 0 ∘ ≤ θ ≤ 90 ∘ and find the angle θ that maximizes the area A . Also find the maximum area.
Constructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets 12 inches wide. After marking off a length of 4 inches from each edge, the builder bends this length up at an angle
. See the illustration. The area
of the opening as a function of
is given by
,
(a) In calculus, you will be asked to find the angle that maximizes A by solving the equation
Solve this equation for
.
(b) What is the maximum area
of the opening?
(c) Graph
and find the angle
that maximizes the area
. Also find the maximum area.
1. A bicyclist is riding their bike along the Chicago Lakefront Trail. The velocity (in
feet per second) of the bicyclist is recorded below. Use (a) Simpson's Rule, and (b)
the Trapezoidal Rule to estimate the total distance the bicyclist traveled during the
8-second period.
t
0 2
4 6 8
V
10 15
12 10 16
2. Find the midpoint rule approximation for
(a) n = 4
+5
x²dx using n subintervals.
1° 2
(b) n = 8
36
32
28
36
32
28
24
24
20
20
16
16
12
8-
4
1
2
3
4
5
6
12
8
4
1
2
3
4
5
6
=
5 37
A 4 8 0.5
06
9
Consider the following system of equations, Ax=b :
x+2y+3z - w = 2
2x4z2w = 3
-x+6y+17z7w = 0
-9x-2y+13z7w = -14
a. Find the solution to the system. Write it as a parametric equation. You can use a
computer to do the row reduction.
b. What is a geometric description of the solution? Explain how you know.
c. Write the solution in vector form?
d. What is the solution to the homogeneous system, Ax=0?
University Calculus: Early Transcendentals (4th Edition)
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