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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.2.24c
Chapter 7, Problem 7.2.24c

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.


c. Find a function that gives the total energy used (in MW-yr) between t=0 and any future time t>0.

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1
Identify the given information: the initial rate of electricity consumption at time \(t=0\) is 2000 MW, and the rate increases by 1.3% per year. This suggests the rate function grows exponentially.
Express the rate of electricity consumption as a function of time \(t\). Since the rate increases by 1.3% per year, the rate function can be written as \(R(t) = 2000 \times (1.013)^t\), where \(t\) is in years.
Understand that the total energy used between \(t=0\) and any future time \(t\) is the integral of the rate function over that interval. So, the total energy function \(E(t)\) is given by the integral \(E(t) = \int_0^t R(x) \, dx\).
Set up the integral explicitly: \(E(t) = \int_0^t 2000 \times (1.013)^x \, dx\). This integral will give the total energy consumed in megawatt-years (MW-yr) from time 0 to time \(t\).
To solve the integral, recall that the integral of an exponential function \(a^x\) with respect to \(x\) is \(\frac{a^x}{\ln(a)}\). Use this formula to express \(E(t)\) in terms of \(t\) without calculating the final numeric value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Growth

Exponential growth describes a process where a quantity increases at a rate proportional to its current value. In this problem, the electricity usage rate grows by 1.3% per year, meaning the rate function can be modeled as an exponential function of time.
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Integration of a Rate Function

To find the total quantity accumulated over time from a rate function, we integrate the rate with respect to time. Here, integrating the power consumption rate over the interval [0, t] gives the total energy used in megawatt-years.
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Units and Interpretation of the Integral

Understanding units is crucial: the rate is in megawatts (MW), and integrating over time (years) yields energy in megawatt-years (MW-yr). This helps interpret the integral as total energy consumed, not just instantaneous power.
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Related Practice
Textbook Question

Chemotherapy In an experimental study at Dartmouth College, mice with tumors were treated with the chemotherapeutic drug Cisplatin. Before treatment, the tumors consisted entirely of clonogenic cells that divide rapidly, causing the tumors to double in size every 2.9 days. Immediately after treatment, 99% of the cells in the tumor became quiescent cells which do not divide and lose 50% of their volume every 5.7 days. For a particular mouse, assume the tumor size is 0.5 cm³ at the time of treatment.

d. Plot a graph of V(t) for 0 ≤ t ≤ 15. What happens to the size of the tumor, assuming there are no follow-up treatments with Cisplatin?

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. ln(1 + √2) = −ln(−1 + √2)

Textbook Question

Terminal velocity Refer to Exercises 95 and 96.


d. How tall must a cliff be so that the BASE jumper (m = 75 kg and k = 0.2) reaches 95% of terminal velocity? Assume the jumper needs at least 300 m at the end of free fall to deploy the chute and land safely.

Textbook Question

ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.

c. Show that ln 2ⁿ > n/2 and ln 2^(−n) < −n/2.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


d. If the rate constant of an exponential growth function is increased, its doubling time is decreased.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.


c. ln (x + y) = ln x + ln y