A large weather balloon is tethered by two ropes. One rope measures 23 ft and attaches to the balloon at an angle of 32 ° from the ground. The second rope attaches to the base of the balloon at an angle of 15 ° with the ground. a. How far from the ground is the balloon floating? Round to the nearest tenth of a foot. b. Find the length of the second rope. Round to the nearest tenth of a foot c. If both ropes suddenly detach and the balloon rises straight up at a rate of 3 ft/ sec , how long will it take the balloon to reach a height of 50 ft from the ground? Round to the nearest tenth of a second.
A large weather balloon is tethered by two ropes. One rope measures 23 ft and attaches to the balloon at an angle of 32 ° from the ground. The second rope attaches to the base of the balloon at an angle of 15 ° with the ground. a. How far from the ground is the balloon floating? Round to the nearest tenth of a foot. b. Find the length of the second rope. Round to the nearest tenth of a foot c. If both ropes suddenly detach and the balloon rises straight up at a rate of 3 ft/ sec , how long will it take the balloon to reach a height of 50 ft from the ground? Round to the nearest tenth of a second.
Solution Summary: The author calculates how far from the ground a floating balloon is tethered by two ropes. One rope measures 23ft and attaches to the balloon at an angle of 32
A large weather balloon is tethered by two ropes. One rope measures
23
ft
and attaches to the balloon at an angle of
32
°
from the ground. The second rope attaches to the base of the balloon at an angle of
15
°
with the ground.
a. How far from the ground is the balloon floating? Round to the nearest tenth of a foot.
b. Find the length of the second rope. Round to the nearest tenth of a foot
c. If both ropes suddenly detach and the balloon rises straight up at a rate of
3
ft/
sec
, how long will it take the balloon to reach a height of
50
ft
from the ground? Round to the nearest tenth of a second.
Consider the function f(x) = x²-1.
(a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative.
Show all your steps clearly.
(b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the
graph where x 1 and x->
1+h (for a small positive value of h, illustrate conceptually). Then,
draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the
value you found in part (a).
(c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in
the context of the graph of f(x). How does the rate of change of this function vary at different
points?
1. The graph of ƒ is given. Use the graph to evaluate each of the following values. If a value does not exist,
state that fact.
и
(a) f'(-5)
(b) f'(-3)
(c) f'(0)
(d) f'(5)
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g'(5)
=
4.
-
3. If an equation of the tangent line to the graph of y = f(x) at the point where x 2 is y = 4x — 5, find ƒ(2)
and f'(2).
College Algebra with Modeling & Visualization (5th Edition)
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