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Two airplanes are flying in the air at the same height: airplane A is flying east at 250 mi/h and airplane B is flying north at 300 mi/h . If they are both heading to the same airport, located 30 miles east of airplane A and 40 miles north of airplane B , at what rate is the distance between the airplanes changing?

A right triangle is formed by two airplanes A and B moving perpendicularly to each other. The hypotenuse is the distance between planes A and B. The other sides are extensions of each plane’s path until they meet.

The distance is decreasing at 390 mi/h .

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You and a friend are riding your bikes to a restaurant that you think is east; your friend thinks the restaurant is north. You both leave from the same point, with you riding at 16 mph east and your friend riding 12 mph north. After you traveled 4 mi, at what rate is the distance between you changing?

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Two buses are driving along parallel freeways that are 5 mi apart, one heading east and the other heading west. Assuming that each bus drives a constant 55 mph, find the rate at which the distance between the buses is changing when they are 13 mi apart, heading toward each other.

The distance between them shrinks at a rate of 1320 13 101.5 mph .

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A 6-ft-tall person walks away from a 10-ft lamppost at a constant rate of 3 ft/sec . What is the rate that the tip of the shadow moves away from the pole when the person is 10 ft away from the pole?

A lamppost is shown that is 10 ft high. To its right, there is a person who is 6 ft tall. There is a line from the top of the lamppost that touches the top of the person’s head and then continues to the ground. The length from the end of this line to where the lamppost touches the ground is 10 + x. The distance from the lamppost to the person on the ground is 10, and the distance from the person to the end of the line is x.
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Using the previous problem, what is the rate at which the tip of the shadow moves away from the person when the person is 10 ft from the pole?

9 2 ft/sec

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A 5-ft-tall person walks toward a wall at a rate of 2 ft/sec. A spotlight is located on the ground 40 ft from the wall. How fast does the height of the person’s shadow on the wall change when the person is 10 ft from the wall?

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Using the previous problem, what is the rate at which the shadow changes when the person is 10 ft from the wall, if the person is walking away from the wall at a rate of 2 ft/sec?

It grows at a rate 4 9 ft/sec

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A helicopter starting on the ground is rising directly into the air at a rate of 25 ft/sec. You are running on the ground starting directly under the helicopter at a rate of 10 ft/sec. Find the rate of change of the distance between the helicopter and yourself after 5 sec.

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Using the previous problem, what is the rate at which the distance between you and the helicopter is changing when the helicopter has risen to a height of 60 ft in the air, assuming that, initially, it was 30 ft above you?

The distance is increasing at ( 135 26 ) 26 ft/sec

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For the following exercises, draw and label diagrams to help solve the related-rates problems.

The side of a cube increases at a rate of 1 2 m/sec. Find the rate at which the volume of the cube increases when the side of the cube is 4 m.

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The volume of a cube decreases at a rate of 10 m/sec. Find the rate at which the side of the cube changes when the side of the cube is 2 m.

5 6 m/sec

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The radius of a circle increases at a rate of 2 m/sec. Find the rate at which the area of the circle increases when the radius is 5 m.

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The radius of a sphere decreases at a rate of 3 m/sec. Find the rate at which the surface area decreases when the radius is 10 m.

240 π m 2 /sec

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The radius of a sphere increases at a rate of 1 m/sec. Find the rate at which the volume increases when the radius is 20 m.

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Questions & Answers

Find the derivative of g(x)=−3.
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evaluate the following computation (x³-8/x-2)
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teach me how to solve the first law of calculus.
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what is differentiation
Ibrahim Reply
f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)
Naufal Reply
fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5
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how to use fundamental theorem to solve exponential
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find the bounded area of the parabola y^2=4x and y=16x
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what is absolute value means?
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(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
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find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
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Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
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log tan (x/4+x/2)
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y=(x^2 + 3x).(eipix)
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A Function F(X)=Sinx+cosx is odd or even?
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f(x)=1/1+x^2 |=[-3,1]
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Practice Key Terms 1

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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