Answer:
(b) is true
Step-by-step explanation:
Given
Molly
[tex]a = 500[/tex] --- starting balance
[tex]m = 10[/tex] --- monthly rate
Her brother
[tex]a = 100[/tex] ---- starting balance
[tex]r = 10\%[/tex] --- annual rate
Required
Determine which option is true
First, we calculate her brother's function.
The function is an exponential function calculated as:
[tex]y = ab^x[/tex]
Where [tex]b = 1 + r[/tex]
So, we have:
[tex]y = ab^x[/tex]
[tex]y = 100 *(1 + 10\%) ^x[/tex]
[tex]y = 100 *(1 + 0.10) ^x[/tex]
[tex]y = 100 *(1.10) ^x[/tex]
Hence:
[tex]g(x) = 100 *(1.10) ^x[/tex]
Next, we calculate Molly's function (a linear function)
The monthly function is:
[tex]y = mx + a[/tex]
So, we have:
[tex]y = 10x + 500[/tex]
Annually, the function will be:
[tex]y = 10x*12 + 500[/tex]
[tex]y = 120x + 500[/tex]
So, we have:
[tex]f(x) = 120x + 500[/tex]
At this point, we have:
[tex]f(x) = 120x + 500[/tex] ---- Molly
[tex]g(x) = 100 *(1.10) ^x[/tex] ---- Her brother
Next, we test each option
(a): Molly's account will have a faster rate of change over [32,40]
We calculated Molly's function to be:
[tex]y = 120x + 500[/tex]
The slope of a linear function with the form: [tex]y = mx + b[/tex] is m
By comparison:
[tex]m = 120[/tex]
Since Molly's account is a linear function, the rate of change over any interval will always be the same; i.e.
[tex]m = 120[/tex]
For his brother:
Rate of change is calculated using:
[tex]m = \frac{g(b) - g(a)}{b - a}[/tex]
[tex]m = \frac{g(40) - g(32)}{40 - 32}[/tex]
[tex]m = \frac{g(40) - g(32)}{8}[/tex]
Calculate g(40) and g(32)
[tex]g(x) = 100 *(1.10) ^x[/tex]
[tex]g(40) = 100 * 1.10^{40} =4526[/tex]
[tex]g(32) = 100 * 1.10^{32} = 2111[/tex]
So, we have:
[tex]m = \frac{4526 - 2111}{8}[/tex]
[tex]m = \frac{2415}{8}[/tex]
[tex]m = 302[/tex]
By comparison: [tex]302 > 120[/tex]
Hence, her brother's account has a faster rate over [32,40]
(a) is false
(b): Molly's account will have a slower rate of change over [24,30]
[tex]m = 120[/tex] --- Molly's rate of change
For his brother:
[tex]m = \frac{g(b) - g(a)}{b - a}[/tex]
[tex]m = \frac{g(30) - g(24)}{30 - 24}[/tex]
[tex]m = \frac{g(30) - g(24)}{6}[/tex]
Calculate g(30) and g(24)
[tex]g(x) = 100 *(1.10) ^x[/tex]
[tex]g(40) = 100 * 1.10^{30} =1745[/tex]
[tex]g(32) = 100 * 1.10^{24} = 985[/tex]
So, we have:
[tex]m = \frac{g(30) - g(24)}{6}[/tex]
[tex]m = \frac{1745 - 985}{6}[/tex]
[tex]m = \frac{760}{6}[/tex]
[tex]m = 127[/tex]
By comparison: [tex]127 > 120[/tex]
Hence, Molly's account has a slower rate over [24,30]
(b) is false
(c): Molly's account will have a slower rate of change over [0,4]
[tex]m = 120[/tex] --- Molly's rate of change
For his brother:
[tex]m = \frac{g(b) - g(a)}{b - a}[/tex]
[tex]m = \frac{g(4) - g(0)}{4 - 0}[/tex]
[tex]m = \frac{g(4) - g(0)}{4}[/tex]
Calculate g(4) and g(0)
[tex]g(x) = 100 *(1.10) ^x[/tex]
[tex]g(4) = 100 * 1.10^4 =146[/tex]
[tex]g(0) = 100 * 1.10^{0} = 100[/tex]
So, we have:
[tex]m = \frac{g(4) - g(0)}{4}[/tex]
[tex]m = \frac{146 - 100}{4}[/tex]
[tex]m = \frac{46}{4}[/tex]
[tex]m = 11.5[/tex]
By comparison: [tex]120>11.5[/tex]
Hence, Molly's account has a faster rate over [0,4]
(c) is false
