Problem 1 :
In January, the parks and Wildlife department released 267 bass into a newly constructed pond. Each month, the population of bass in the pond increases by 4.2%. At this rate of growth, what function can be used to determine the population of bass m months after January ?
Solution :
The initial amount of bass = 267
Each month the population of bass increases by 4.2%.
Initial population be 100%.
Growth rate = 100% + 4.2%
= 104.2%
= 1.042
Required exponential function :
y = 267 (1.042)^{m}
Problem 2 :
A pyramid at chicken Itza in Mexico contains several layers that are in the shape of square prisms. The first layer has a side of 50 meters and each successive layer has a side length that is 95% of the one directly below it. What function can be used to find the side length, in meters of layer, where 1 ≤ L ≤ 6 ?
Solution :
Initial value = 50 meter, a = 50
Side length of every layer decreases by 95%.
y = 50(0.95)^{L}
Here L = 1, 2, 3, 4, 5, 6
Problem 3 :
An official NBA basketball must be inflated so that when it is dropped, it must not bounce back more than 75% of he height from which it was dropped. If a basket ball is dropped from a height of 72 inches, what function r(t) describes the height of the basket ball as a function of t, the number of bounces ?
Solution :
Initial height when the basket ball is dropped (a) = 72 inches
Every time it bounce back to 75% of height from which it was dropped
Common ratio = (100 + 75)%
= 175%
b = 1.75
Number of times bouncing back = n
r(t) = 72 (1.75)^{t}
Problem 4 :
The population of Bexar Country Texas, for a few recent years is shown in the table.
If t represents the number of years since 2010 and P(t) represents the population in millions, what function, P(t) best describes the data in the table ?
Solution :
Initial value(a) = 1.723
Check if the multiplication factor are same :
1.756/1.723 ==> 1.01
1.789/1.756 ==> 1.01
1.822/1.789 ==> 1.01
1.856/1.822 ==> 1.01
Common ratio is same, so it is exponential function.
b = 1.01
P(t) = 1.723 (1.01)^{t}
Problem 5 :
A chiropterologist is a scientist who studies bats. A bat colony is discovered and a chiropterologist calculates that there are 2100 bats in the colony. If the population of the colony doubles each year, which function, B(t) describes the population of the colony t year after discovery ?
(a) B(t) = 2100(1/2)^{t }(b) B(t) = 2100(2)^{t}
(c) B(t) = 2100 +2^{t }(d) B(t) = 2100 + (1/2)^{t}
Solution :
Initial population(a) = 2100
In the second year, population of bats = 2(2100)
In the third year, population of bats = 2[2(2100)] => 2^{2} (2100)
After t years, the population :
B(t) = 2^{t} (2100)
Problem 6 :
Chuy purchased a used truck for $11500. According to an online vehicle website, his truck will depreciate or lose value, at a rate of 5.5% each year. What function d(x) represents the value of Chuy's truck x year after its purchase ?
(a) d(x) = 11500(0.945)^{x }(b) d(x) = 11500(1.055)^{x}
(c) d(x) = 11500(0.055)^{x }(d) d(x) = 11500(5.5)^{x}
Solution :
Initial value a = 11500
Since the value is depreciating, common ratio = (100-5.5)%
= 94.5%
= 0.945
x be the number of years after its purchase.
d(x) = 11500 (0.945)^{x}
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM