5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 16:5

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

Step 3: Substitutek=

1

2

ln

(

8

5

)

into one of the equations.

500 =y 0 ek

500 =y 0 e

ln

(√ 8

5

)

500 =y 0

(√

8

5

)

y 0 =

500

√

8 / 5

= 125

√

10 ≈ 395. 285

Thus, there are 395 bacteria present initially.

(b) y 0 = 125

√

10,k=ln

√

8

5

y(t)=y 0 ekt

y(t)=

(

125

√

10

)

e

(

ln

√ 8

5

)t

=

(

125

√

10

)( 8

5

)(1/2)t

y( 4 )=

(

125

√

10

)( 8

5

)(1/2)4

=

(

125

√

10

)( 8

5

) 2

= 1011. 93

Thus, there are approximately 1012 bacteria present after 4 days.

TIP • Get a good night’s sleep the night before. Have a light breakfast before the exam.

Example 2 Radioactive Decay

Carbon-14 has a half-life of 5750 years. If initially there are 60 grams of carbon-14, how

many grams are left after 3000 years?

Step 1: y(t)=y 0 ekt= 60 ekt

Since half-life is 5750 years, 30= 60 ek(5750)⇒

1

2

=e^5750 k.

ln

(

1

2

)

=ln

(

e^5750 k

)

−ln 2= 5750 k

−ln 2

5750

=k