A company is planning to expand its business in a few years. New plant construction costs are estimated to be $4.75 per square foot. The company invests $850,000 today at 6% compounded quarterly.
a. How many square feet could be built in 3 1/2 years?
b. If the company waits for a total of five years, the construction costs increase to $5.10 per square foot. How many square feet could they build if they waited a total of five years?
850000(1+.06/4)^(4*3.5)/4.75 = 220,419
850000(1+.06/4)^(4*5)/5.10 = 224,476
To solve these problems, we need to use the compound interest formula:
A = P * (1 + (r/n))^(nt)
Where:
A = Final amount
P = Principal amount (initial investment)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years
a. How many square feet could be built in 3 1/2 years?
Given:
Initial investment (P) = $850,000
Annual interest rate (r) = 6% or 0.06
Compounded quarterly (n) = 4
Time (t) = 3.5 years
Using the compound interest formula:
A = P * (1 + (r/n))^(nt)
A = $850,000 * (1 + (0.06/4))^(4 * 3.5)
Calculating the expression inside the parentheses first:
(0.06/4) = 0.015
(1 + 0.015)^(4 * 3.5) = 1.015^14
Calculating the value of 1.015^14:
1.015^14 = 1.233354811
Now, let's calculate the final amount (A):
A = $850,000 * 1.233354811
A ≈ $1,048,351.58
The final amount represents the total funds available for construction.
To calculate the number of square feet that could be built using the given construction cost per square foot ($4.75), we need to divide the final amount by the cost per square foot:
Square feet = A / cost per square foot
Square feet = $1,048,351.58 / $4.75
Square feet ≈ 220,655.08
Therefore, approximately 220,655 square feet could be built in 3 1/2 years.
b. If the company waits for a total of five years, the construction costs increase to $5.10 per square foot. How many square feet could they build if they waited a total of five years?
Using the same formula, and given construction cost per square foot ($5.10), we now calculate the final amount (A) after five years:
A = $850,000 * (1 + (0.06/4))^(4 * 5)
Calculating the expression inside the parentheses first:
(0.06/4) = 0.015
(1 + 0.015)^(4 * 5) = 1.015^20
Calculating the value of 1.015^20:
1.015^20 = 1.34897254
Now, let's calculate the final amount (A):
A = $850,000 * 1.34897254
A ≈ $1,146,626.66
To calculate the number of square feet that could be built using the given construction cost per square foot ($5.10), we need to divide the final amount by the cost per square foot:
Square feet = A / cost per square foot
Square feet = $1,146,626.66 / $5.10
Square feet ≈ 224,927.21
Therefore, approximately 224,927 square feet could be built if they waited a total of five years.
To solve these problems, we need to use the compound interest formula to calculate the future value of the investment. Then, we can use this value to determine how many square feet the company can build.
The compound interest formula can be written as:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal investment amount (initial amount)
r = annual interest rate (expressed as a decimal)
n = number of times that interest is compounded per year
t = time (in years)
a. To find out how many square feet can be built in 3 1/2 years, we can use the compound interest formula.
Given:
P = $850,000
r = 6% = 0.06
n = 4 (compounded quarterly)
t = 3.5 years
Calculate A using the compound interest formula:
A = $850,000(1 + 0.06/4)^(4*3.5)
Simplify the equation:
A = $850,000(1.015)^14
Use a calculator to find the value of A.
A ≈ $850,000 * 1.2317055
So, the future value of the investment after 3 1/2 years is approximately $1,047,449.73.
Now, we divide the future value by the construction cost per square foot to determine the number of square feet that can be built:
Square feet = Future value / Construction cost per square foot
Square feet = $1,047,449.73 / $4.75
Square feet ≈ 220,617.28
Therefore, the company can build approximately 220,617 square feet in 3 1/2 years.
b. If the company waits a total of five years, the construction costs increase to $5.10 per square foot.
Given:
P = $850,000
r = 6% = 0.06
n = 4 (compounded quarterly)
t = 5 years
Calculate A using the compound interest formula:
A = $850,000(1 + 0.06/4)^(4*5)
Simplify the equation:
A = $850,000(1.015)^20
Use a calculator to find the value of A.
A ≈ $850,000 * 1.3498588
So, the future value of the investment after 5 years is approximately $1,147,380.94.
Now, we can divide the future value by the new construction cost per square foot to determine the number of square feet that can be built:
Square feet = Future value / Construction cost per square foot
Square feet = $1,147,380.94 / $5.10
Square feet ≈ 224,903.13
Therefore, if the company waits a total of five years, they can build approximately 224,903 square feet.