A construction company is building a new parking garage and has established the following rates: $5,000 per month for the initial three months, $8,000 per month for the next three months, and a one-time payment of $5,000 for the final four months upon completion. The lump sum for the last period is due at the conclusion of the sixth month. Let the cost C (in thousands of dollars) be a function of the time t (in months), describing the amount payable as the company progresses on the parking garage project. The cost function C(t) is defined as: For 0 < t ≤ 3: C(t) = 5t, reflecting the charge of $5,000 per month for the first three months. For 3 < t ≤ 6: C(t) = 15 + 8(t - 3), taking into account the accrued $15,000 from the first three months and $8,000 per month for the following three months. For t > 6: C(t) = 44, accounting for the total of $15,000 and $24,000 from the first six months and the final $5,000 for the last four months. To summarize the piecewise function: C(t) = { 5t for 0 < t ≤ 3; 15 + 8(t - 3) for 3 < t ≤ 6; 44 for t > 6. } Note: This cost function presumes that rates change precisely after three months and that the lump sum is paid exactly at the end of the sixth month. If payment schedules or project timelines alter, modifications to the function would be necessary.

Question

A construction company is building a new parking garage and has established the following rates: $5,000 per month for the initial three months, $8,000 per month for the next three months, and a one-time payment of $5,000 for the final four months upon completion. The lump sum for the last period is due at the conclusion of the sixth month. Let the cost C (in thousands of dollars) be a function of the time t (in months), describing the amount payable as the company progresses on the parking garage project. The cost function C(t) is defined as: For 0 < t ≤ 3: C(t) = 5t, reflecting the charge of $5,000 per month for the first three months. For 3 < t ≤ 6: C(t) = 15 + 8(t - 3), taking into account the accrued $15,000 from the first three months and $8,000 per month for the following three months. For t > 6: C(t) = 44, accounting for the total of $15,000 and $24,000 from the first six months and the final $5,000 for the last four months. To summarize the piecewise function: C(t) = { 5t for 0 < t ≤ 3; 15 + 8(t - 3) for 3 < t ≤ 6; 44 for t > 6. } Note: This cost function presumes that rates change precisely after three months and that the lump sum is paid exactly at the end of the sixth month. If payment schedules or project timelines alter, modifications to the function would be necessary.

Mathematics · College · Thu Feb 04 2021

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Answered on Mon Nov 20 2023

The cost function C(t) represents the total amount due to the construction company as they build the parking garage over time. From the given rates, we can write the cost function as a piecewise function as follows:

C(t) = { 5t, for 0 < t ≤ 3; 15 + 8(t - 3), for 3 < t ≤ 6; 44, for t > 6. }

This means: - For the time interval from the start of work up to and including the third month, the cost is calculated as $5,000 times the number of months, hence the term 5t where t is the current month. - After the third month and up to the sixth month, the cost is calculated as $15,000 (the total from the first three months) plus $8,000 per month for the months beyond the third. Thus, for this period, the function is 15 + 8(t - 3), where (t - 3) calculates the number of months beyond the third month. - Finally, after the sixth month, the cost is fixed at a total amount due of $44,000 because $15,000 is for the first three months, $24,000 is for the fourth to the sixth months [$8,000/month × 3 months], and $5,000 is the lump sum for the last four months.

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