Antonia is keeping track of the number of pages she reads. She started the school year having already read 95 pages of her book, and she then reads 20 pages each day. The function for her total pages read is f(x) = 20x + 95. What do f(x) and x represent in Antonia's situation?
f(x) represents the number of days; x represents the total number of pages Antonia has read.
f(x) represents the total number of pages Antonia has read; x represents the number of days.
f(x) represents the number of pages Antonia reads per day; x represents the number of days.
f(x) represents the total number of pages Antonia has read; x represents the number of pages she reads per day.
f(x) represents the total number of pages Antonia has read; x represents the number of days.
The balances in two separate bank accounts that grow each month at different rates are represented by the functions f(x) and g(x). In what month do the funds in the f(x) bank account exceed those in the g(x) bank account?
Month (x) f(x) = 2x g(x) = 4x + 12
1 2 16
2 4 20
Month 3
Month 4
Month 5
Month 6
To determine in which month the funds in the f(x) bank account exceed those in the g(x) bank account, we need to find the value of x for which f(x) > g(x).
f(x) = 2x
g(x) = 4x + 12
Setting them equal, we have:
2x = 4x + 12
Simplifying:
-2x = 12
x = -6
Since we are dealing with months, we exclude negative values. Thus, we start from month 1 and check when f(x) exceeds g(x).
For x = 1:
f(1) = 2(1) = 2
g(1) = 4(1) + 12 = 16
For x = 2:
f(2) = 2(2) = 4
g(2) = 4(2) + 12 = 20
For x = 3:
f(3) = 2(3) = 6
g(3) = 4(3) + 12 = 24
At month 4, f(x) = 2(4) = 8, while g(x) = 4(4) + 12 = 28. Since f(x) (8) exceeds g(x) (28) in month 4, the funds in the f(x) bank account exceed those in the g(x) bank account in month 4.
The amount of a radioactive material changes with time. The table below shows the amount of radioactive material f(t) left after time t:
t(hours) 0 1 2
f(t) 180 90 45
Which exponential function best represents the relationship between f(t) and t?
f(t) = 0.5(180)t
f(t) = 180(0.25)t
f(t) = 180(0.5)t
f(t) = 0.5(50)t
To find the best exponential function that represents the relationship between f(t) and t, we need to determine the common ratio (r) in the exponential growth formula f(t) = a * r^t.
By observing the given data, we can see that dividing a term by its previous term results in the same value each time:
90/180 = 0.5
45/90 = 0.5
This implies that the common ratio (r) in the exponential growth formula is 0.5.
Now we just need to find the initial amount (a). Looking at the table, when t = 0, f(t) = 180. This means that a * r^0 = a = 180.
Therefore, the exponential function that best represents the relationship between f(t) and t is f(t) = 180(0.5)^t.
Bob opened a new furniture store, and his daily sales are modeled by f(x) = 1000(1.15)x. Determine the rate of growth.
1000%
115%
15%
85%
The rate of growth is determined by the coefficient of the exponential function. In this case, the coefficient is 1.15.
To determine the rate of growth, we need to subtract 1 from the coefficient and multiply by 100 to express it as a percentage.
(1.15 - 1) * 100 = 0.15 * 100 = 15%
Therefore, the rate of growth for Bob's daily sales is 15%.
The value of a rare painting has increased each year since it was found at a garage sale. The value of the painting is modeled by the function f(x) = 799(1.03)x. What does the 799 represent? What will the painting be worth after five years? Round your answer to the nearest dollar.
799 represents the value of the painting when it was found; the painting will be worth $926 after five years.
799 represents the increase in the value of the painting; the painting will be worth $103 after five years.
799 represents the value of the painting when it was found; the painting will be worth $804 after five years.
799 represents the total value of the painting; the painting will be worth $926 after five years.
799 represents the value of the painting when it was found.
To find the painting's worth after five years, we substitute x = 5 into the function:
f(5) = 799(1.03)^5 ≈ $925.96.
Rounded to the nearest dollar, the painting will be worth $926 after five years.
Therefore, the correct answer is:
799 represents the value of the painting when it was found; the painting will be worth $926 after five years.
Enrollment at a golf academy has grown exponentially since the academy opened. A graph depicting this growth is shown. Determine the percentage rate of growth.
An exponential graph has Time in Years on the x axis and Enrollments on the y axis. A curve that rises from left to right begins at zero comma fifteen and passes through four comma seventy five.
1.5%
5%
50%
0.5%