a Determine the domain and range considering the minimum and maximum values of t and P.
B
b Substitute t=12 000 in the function.
C
c Write the a function for the new amount and compare the functions.
D
d Does the vertical stretch change the minimum and maximum values of t and P for the given case?
A
aDomain: 0 < t
Range: 0< P < 100
B
b About 69.77 grams
C
c A vertical translation by a factor of 5.5
D
d Yes, see solution
Practice makes perfect
a Let's analyze the given function.
P=100(0.99997)^t
We know that t represents the time and P represents the weight of the element. Therefore, either of them cannot be negative. Moreover, because the initial amount of plutonium-239 is 100 grams and the given function is an exponenetial decay, the range cannot be greater than 100. With this, the domain and range can be stated as below.
Domain:& 0 < t
Range:& 0< P < 100
b To find the amount of plutonium-239 after 12 000 years, we will substitute t=12 000 in the function. Let's do it!
Therefore, the amount of plutonium-239 is about 69.77 grams.
c Let's first write the function for 550 grams plutonium-239.
P_(100)=100(0.99997)^t
P_(550)=550(0.99997)^t
The new amount is 5.5 time the old amount. To write the function for 550 grams plutonium-239, the output P_(100) needs to be multiplied by a constant greater than 1.
P_(550)=5.5P_(100)
Therefore, the transformation is a vertical stretch by a factor of 5.5.
d Because the vertical stretch by a factor of 5.5 changes the initial amount from 100 grams to 550 grams, the maximum value of the range also changes.
0< P < 100 ⇒ 0< P < 550
Therefore, the vertical stretch affects the range but it does not affect the domain.
