Problem 1 :
The number of dandelions in a large park is recorded over the course of five months, as shown in the table below.
Month 1 2 3 4 5 |
Dandelions 12500 2500 500 100 20 |
which of the following best describes the relationship between time and the number of dandelions during the five months ?
a) Increasing linear b) Decreasing linear
c) Exponential growth d) Exponential decay
Solution :
Considering the outputs, multiplying each output by the ratio 1/5, we will get the next output.
Multiplication factor (b) lies between 0 to 1. So, it is exponential decay.
Problem 2 :
The number of subscribers S, to a magazine increases by 21 percent each year. If the current number of subscribers to the magazine is 3000, which of the following equation models the number of subscribers to the magazine h half years from now ?
a) S = 3000(1.1)^{h} b) S = 3000(1.21)^{h}
c) S = 3000(1.01)^{h} d) S = 3000(1.4641)^{h}
Solution :
Increase = 21%
Number of subscribers = 3000
Exponential growth function :
y = a(1+r%)^{x}
x = h = number of half years
y = S = number of subscribers
and r = 21%
S = a(1+r%)^{h}
S = 3000(1+21%)^{h}
S = 3000(1.21)^{h}
Problem 3 :
The total amount of water w, in gallons left in a tank can be modeled by the equation w = 300 - 5t, where t is the number of hours since the tank started leaking. Which of the following is the best interpretation of the number 5 in the equation ?
a) The tank is empty after 5 hours.
b) The tank loses 5 gallons of water each hour.
c) the tank continues to lost water until 5 gallons are left.
d) Each hour, the tank loses 5 less gallons of water than it did the previous hour.
Solution :
The given function w is linear decreasing function. By comparing with y = mx + b
Rate of change (b) = -5
The tank losses 5 gallons of water each hour. So, option b is correct.
Problem 4 :
Tom buys a pack of baseball cards everyday. Each pack contains 7 cards but he gives away the two least valuable ones to his brother. Which of the following best describes the relationship between time (in days) and the total number of baseball cards in Tom's collection ?
a) Increasing linear b) Decreasing linear
c) Exponential growth d) Exponential decay
Solution :
Each day he buys a pack containing 7 cards.
Number of cards giving to his brother = 2
7 - 2 = 5
There is a constant increase, then increasing linear.
Problem 5 :
Anna opens a bank account with an initial deposit of $1000. The bank account will earn 3 percent interest compounded annually for the first 5 years, after which it will earn 7 percent interest compounded annually. Which of the following expressions represents the total amount in the account after t years, where t > 5 ?
a) 1000 (1.03)^{5} (1.07)^{t }b) 1000 (1.03)^{t-5 }(1.07)^{t}
c) 1000 (1.03)^{5} (1.07)^{t-5 }d) 1000 (1.03)^{5 }(1.07)^{t+5 }
Solution :
Initial deposit = 1000
3% compounded annually
Total number of years investing = t.
= 1000 (1 + 3%)^{5} (1 + 7%)^{t-5}
= 1000 (1 + 3/100)^{5} (1 + 7/100)^{t-5}
= 1000 (1 + 0.03)^{5} (1 + 0.07)^{t-5}
= 1000 (1.03)^{5} (1.07)^{t-5}
Problem 6 :
A house is losing a fourth of its value every year. Which of the following best describes the relationship between time (in years) and the value of house ?
a) Increasing linear b) Decreasing linear
c) Exponential growth d) Exponential decay
Solution :
Let x be the value of house and it be a 100%.
Value of house after 1 year = 75% of x
= 0.75x
Value of house after 2 years = 75% of (75% of x),..............
= 0.75(0.75x)
= x(0.75)^{2}
Exponential decay.
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