Answer:
$24,446
Step-by-step explanation:
To find the average annual growth rate, we can use the formula for exponential growth:
[tex]\Large\boxed{\boxed{ A = P(1 + r)^n }}[/tex]
Where:
We already know the initial amount ([tex] P = 14,000 [/tex]), the final amount ([tex] A = 18,500 [/tex]), and the number of years ([tex] n = 4 [/tex]).
Let's find the annual growth rate ([tex] r [/tex]) first:
[tex] 18,500 = 14,000(1 + r)^4 [/tex]
Divide both sides by 14,000:
[tex]\dfrac{18,500}{14,000} = (1 + r)^4[/tex]
[tex] 1.3214285714285 = (1 + r)^4 [/tex]
Now, we take the fourth root of both sides:
[tex] (1 + r) = \sqrt[4]{1.3214285714285} [/tex]
[tex] 1 + r = 1.072163265 [/tex]
Subtract 1 from both sides:
[tex] r = 1.072163265 - 1 [/tex]
[tex] r = 0.07216326513 [/tex]
To convert in percentage, multiply by 100.
[tex] r = 0.07216326513 \cdot 100 \% [/tex]
[tex] r = 7.216326513 \% [/tex]
[tex] r = 7.216% \textsf{ (in 3 d.p.)} \% [/tex]
So, the average annual growth rate is approximately 7.216 %.
Now that we have the growth rate, we can calculate the expenses four years from now:
We have the initial amount [tex] (P = 18,500) [/tex], [tex] ( r = 7.216\% = 0.07216 )[/tex] .and the number of years ([tex] n = 4 [/tex]).
Substitute the value in above formula, we get:
[tex] P = 18,500(1 + 0.07216)^4 [/tex]
[tex] P = 18,500(1.07216)^4 [/tex]
Calculating:
[tex] P = 18,500 \times 1.07216^4 [/tex]
[tex] P \approx 18,500 \times 1.321412475 [/tex]
[tex] P \approx 24446.13078 [/tex]
[tex] P \approx 24446 \textsf{(in nearest whole number)}[/tex]
So, the expenses four years from now, if the growth rate continues, will be approximately $24,446.
Answer:
$24,446.43
Step-by-step explanation:
To determine the yearly expenses at a State University four years from now, given that the yearly expenses have increased from $14,000 to $18,500 in the last four years, we can calculate the growth rate over this period and then apply it to the current yearly expenses to project future expenses four years from now.
[tex]\textsf{Growth rate $(r)$} = \dfrac{18500-14000}{14000} \times 100\% \\\\\\ \textsf{Growth rate $(r)$} = \dfrac{9}{28} \times 100\% \\\\\\ \textsf{Growth rate $(r)$} = 0.3214285714... \times 100\% \\\\\\ \textsf{Growth rate $(r)$} = 32.14\%[/tex]
Therefore, the growth rate over the last 4 years was approximately 32.14%.
To calculate the expenses four years from now, assuming the growth rate continues, we apply the growth rate (r) to the current yearly expenses of $18,500:
[tex]\textsf{Expenses} = 18500(1+r) \\\\ \textsf{Expenses} = 18500(1+0.3214285714...) \\\\ \textsf{Expenses} = 18500 \times 1.3214285714... \\\\ \textsf{Expenses} = \$24,446.43 \;\textsf{(nearest cent)}[/tex]
Therefore, if the growth rate continues, the expenses four years from now are projected to be $24,446.43 (rounded to the nearest cent).
