5 Steps to a 5 AP Calculus BC 2019

154 STEP 4. Review the Knowledge You Need to Score High

200 m

x

y

θ

Camera

Balloon

Figure 8.1-5

Step 1: Letxbe the distance between the balloon and the ground;θbe the camera’s angle

of elevation; andtbe the time in seconds.

Step 2: Given:

dx

dt

=10 m/sec; distance between camera and the point on the ground

where the balloon took off is 200 m, tanθ=

x

200

.

Step 3: Find

dθ

dt

atx=150 m.

Step 4: Differentiate both sides with respect tot.

sec^2 θ

dθ

dt

=

1

200

dx

dt

;

dθ

dt

=

1

200

(

1

sec^2 θ

)

(10)=

1

20 sec^2 θ

.

Step 5: secθ=

y

200

and atx=150.

Using the Pythagorean Theorem:y^2 =x^2 +(200)^2

y^2 =(150)^2 +(200)^2

y=± 250.

Sincey>0, theny=250. Thus, secθ=

250

200

=

5

4

.

Evaluating

dθ

dt

∣∣

∣

∣x= 150 =

1

20 sec^2 θ

=

1

20

(

5

4

) 2 radian/sec

=

1

20

(

5

4

) 2 =

1

20

(

25

16

)=^1

125

4

=

4

125

radian/sec

or.032 radian/sec

= 1 .833 deg/sec.

Step 6: The camera’s angle of elevation changes at approximately 1.833 deg/sec when the

balloon is 150 m in the air.