Carrying a Ladder around a Corner Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the illustration. It can be shown that the length L of the ladder as a function of θ is L ( θ ) = 4 csc θ + 3 sec θ . a. In calculus, you will be asked to find the length of the longest ladder that can turn the comer by solving the equation 3 sec θ tan θ − 4 csc θ cot θ = 0 0 ∘ < θ < 90 ∘ Solve this equation for θ . b. What is the length of the longest ladder that can be carried around the corner? c. Graph L = L ( θ ) , 0 ∘ ≤ θ ≤ 90 ∘ , and find the angle θ that minimizes the length L . d. Compare the result with the one found in part (a). Explain why the two answers are the same.
Carrying a Ladder around a Corner Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the illustration. It can be shown that the length L of the ladder as a function of θ is L ( θ ) = 4 csc θ + 3 sec θ . a. In calculus, you will be asked to find the length of the longest ladder that can turn the comer by solving the equation 3 sec θ tan θ − 4 csc θ cot θ = 0 0 ∘ < θ < 90 ∘ Solve this equation for θ . b. What is the length of the longest ladder that can be carried around the corner? c. Graph L = L ( θ ) , 0 ∘ ≤ θ ≤ 90 ∘ , and find the angle θ that minimizes the length L . d. Compare the result with the one found in part (a). Explain why the two answers are the same.
Carrying a Ladder around a Corner Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the illustration. It can be shown that the length
of the ladder as a function of
is
.
a. In calculus, you will be asked to find the length of the longest ladder that can turn the comer by solving the equation
Solve this equation for
.
b. What is the length of the longest ladder that can be carried around the corner?
c. Graph
,
, and find the angle
that minimizes the length
.
d. Compare the result with the one found in part (a). Explain why the two answers are the same.
4
In the integral dxf1dy (7)², make the change of variables x = ½(r− s), y = ½(r + s), and
evaluate the integral. Hint: Find the limits on r and s by sketching the area of integration in the (x, y) plane along
with the r and s axes, and then show that the same area can be covered by s from 0 to r and r from 0 to 1.
7. What are all values of 0, for 0≤0<2л, where 2 sin² 0=-sin?
-
5π
6
π
(A) 0, л,
and
6
7π
(B) 0,л,
11π
, and
6
6
π 3π π
(C)
5π
2 2 3
, and
π 3π 2π
(D)
2' 2'3
, and
3
4元
3
1
די
}
I
-2m
3
1
-3
บ
1
#
1
I
3#
3m
8. The graph of g is shown above. Which of the following is an expression for g(x)?
(A) 1+ tan(x)
(B) 1-tan (x)
(C) 1-tan (2x)
(D) 1-tan
+
X
-
9. The function j is given by j(x)=2(sin x)(cos x)-cos x. Solve j(x) = 0 for values of x in the interval
Quiz A: Topic 3.10
Trigonometric Equations and Inequalities
Created by Bryan Passwater
can you solve this question using the right triangle method and explain the steps used along the way
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