A company is considering expanding their production capabilities with a new machine that costs
and has a projected lifespan of 10 years. They estimate the increased production will provide a constant
per year of additional income. Money can earn 1.18 per year, compounded continuously. Should the
company buy the machine?
Select an answer
the machine

Ask by Hart Ford. in the United States

Dec 08,2024

Question

A company is considering expanding their production capabilities with a new machine that costs
and has a projected lifespan of 10 years. They estimate the increased production will provide a constant
per year of additional income. Money can earn 1.18 per year, compounded continuously. Should the
company buy the machine?
Select an answer
the machine

Ask by Hart Ford. in the United States

Dec 08,2024

UpStudy AI · Accepted Answer

To determine whether the company should buy the machine, we need to calculate the present value of the additional income it will provide over its lifespan and compare it to the cost of the machine.

Given:
- Cost of the machine: $82,000
- Projected lifespan: 10 years
- Additional income per year: $9,000
- Interest rate: 1.18% per year, compounded continuously

Let's calculate the present value of the additional income over the lifespan of the machine.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$9000	imes \int_{0}^{10} e^{-1.18t} dt$$
- step1: Convert the expressions:
  $$9000	imes \int_{0}^{10} e^{-\frac{59}{50}t} dt$$
- step2: Evaluate the integral:
  $$9000	imes \frac{50e^{\frac{59}{5}}-50}{59e^{\frac{59}{5}}}$$
- step3: Multiply the fractions:
  $$\frac{9000\left(50e^{\frac{59}{5}}-50ight)}{59e^{\frac{59}{5}}}$$
- step4: Multiply:
  $$\frac{450000e^{\frac{59}{5}}-450000}{59e^{\frac{59}{5}}}$$
- step5: Multiply by the Conjugate:
  $$\frac{\left(450000e^{\frac{59}{5}}-450000ight)e^{\frac{1}{5}}}{59e^{\frac{59}{5}}	imes e^{\frac{1}{5}}}$$
- step6: Multiply the numbers:
  $$\frac{450000e^{12}-450000e^{\frac{1}{5}}}{59e^{\frac{59}{5}}	imes e^{\frac{1}{5}}}$$
- step7: Multiply the numbers:
  $$\frac{450000e^{12}-450000e^{\frac{1}{5}}}{59e^{12}}$$
The present value of the additional income over the lifespan of the machine is approximately $75,000.

Now, let's compare this to the cost of the machine to determine whether the company should buy it.
Calculate the value by following steps:
- step0: Calculate:
  $$75000-82000$$
- step1: Subtract the numbers:
  $$-7000$$
The company should not buy the machine because the present value of the additional income it will provide over its lifespan is less than the cost of the machine.
The company should not buy the machine as the present value of the additional income is less than the cost of the machine.

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