Omega Instruments has budgeted $300,000 per year to pay for certain ce- ramic parts over the next 5 years. If the company expects the cost of the parts to increase uniformly according to an arith- metic gradient of $1 0,000 per year, what is it expecting the cost to be in year 1, if the interest rate is 10% per year?

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When Lisa was first hired by the Environmental Protection Agency as a research scientist,

she had sufficient income to deposit $600 each quarter into an IRA paying 10% interest,
compounded quarterly. The quarterly deposits lasted for 12 years.
a) How much was in the account at the end of 12 years?
b) Because of her parents’ nursing home costs, Lisa was not able to continue these
deposits. Instead, she deposited the entire IRA account into a 30-year certificate of
deposit earning 12% compounded monthly. What was the value of the account when
it matured?

To find the expected cost of the ceramic parts in year 1, we need to calculate the accumulated value of the budgeted amount over 5 years with an arithmetic gradient of $10,000 per year.

Step 1: Calculate the common difference (d) of the arithmetic gradient.
Since the cost is expected to increase uniformly by $10,000 per year, the common difference (d) is $10,000.

Step 2: Calculate the sum (S) of the arithmetic gradient for 5 years.
The sum (S) of an arithmetic gradient can be calculated using the formula:
S = (n/2) * (2a + (n−1)d)
Where n is the number of terms, a is the first term, and d is the common difference.

In this case, the number of terms (n) is 5, the first term (a) is $300,000, and the common difference (d) is $10,000.
S = (5/2) * (2 * $300,000 + (5−1) * $10,000)
S = (5/2) * ($600,000 + 4 * $10,000)
S = (5/2) * ($600,000 + $40,000)
S = (5/2) * $640,000
S = (5 * $640,000) / 2
S = $3,200,000

Step 3: Calculate the accumulated value in year 1 with an interest rate of 10%.
To calculate the accumulated value, we use the formula:
A = P(1 + r)^n
Where A is the accumulated value, P is the principal amount, r is the interest rate, and n is the number of years.

In this case, the principal amount (P) is the sum (S) of $3,200,000, the interest rate (r) is 10% expressed as a decimal (0.10), and the number of years (n) is 1.
A = $3,200,000 * (1 + 0.10)^1
A = $3,200,000 * (1 + 0.10)
A = $3,200,000 * 1.10
A = $3,520,000

Therefore, Omega Instruments is expecting the cost of the ceramic parts to be $3,520,000 in year 1 if the interest rate is 10% per year.