Power and energy are often used interchangeably, but they are quite different. Energy is measured in units of joules ( J ) or Calories (Cal),
where . On the other hand, power is the rate at which energy is used and is measured in watts . Other useful
units of power are kilowatts and megawatts ( ). If energy is used at a rate of 1 kW for 1 hr , the total amount of energy
used is 1 kilowatt-hour , which is . Suppose the power function of a large city over a 24 -hr period is given by
, where P is measured in MW and corresponds to p.m. (see accompanying figure). Complete
parts (a) through (d) below.
Click the icon to view the graph of .
a. How much energy is consumed by this city in a typical 24 -hr period? Express the answer in MWh and in J .
The energy consumed in a typical 24 -hour period in MWh is 8280 MWh .
(Simplify your answer.)
The energy consumed in a typical 24 -hour period in J is
(Simplify your answer. Use scientific notation.) J .

Ask by Stewart Ryan. in the United States

Jan 23,2025

Question

Power and energy are often used interchangeably, but they are quite different. Energy is measured in units of joules ( J ) or Calories (Cal),
where . On the other hand, power is the rate at which energy is used and is measured in watts . Other useful
units of power are kilowatts and megawatts ( ). If energy is used at a rate of 1 kW for 1 hr , the total amount of energy
used is 1 kilowatt-hour , which is . Suppose the power function of a large city over a 24 -hr period is given by
, where P is measured in MW and corresponds to p.m. (see accompanying figure). Complete
parts (a) through (d) below.
Click the icon to view the graph of .
a. How much energy is consumed by this city in a typical 24 -hr period? Express the answer in MWh and in J .
The energy consumed in a typical 24 -hour period in MWh is 8280 MWh .
(Simplify your answer.)
The energy consumed in a typical 24 -hour period in J is
(Simplify your answer. Use scientific notation.) J .

Ask by Stewart Ryan. in the United States

Jan 23,2025

UpStudy AI · Accepted Answer

To find the total energy consumed by the city in a typical 24-hour period, we need to integrate the power function $$ P(t) = 345 - 230 \sin(\pi t / 12) $$ over the time period from 0 to 24 hours.

Given:
$$ P(t) = 345 - 230 \sin(\pi t / 12) $$

The total energy consumed in a 24-hour period is given by the integral of the power function over the time period:
$$ \text{Total Energy} = \int_{0}^{24} P(t) dt $$

Substitute the power function into the integral:
$$ \text{Total Energy} = \int_{0}^{24} (345 - 230 \sin(\pi t / 12)) dt $$

Now, we can calculate the total energy consumed by the city in a typical 24-hour period.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int_{0}^{24} \left(345-230\sin\left(\frac{\pi t}{12}\right)\right) dt$$
- step1: Evaluate the integral:
  $$\int \left(345-230\sin\left(\frac{\pi t}{12}\right)\right) dt$$
- step2: Use properties of integrals:
  $$\int 345 dt+\int -230\sin\left(\frac{\pi t}{12}\right) dt$$
- step3: Evaluate the integral:
  $$345t+\int -230\sin\left(\frac{\pi t}{12}\right) dt$$
- step4: Evaluate the integral:
  $$345t+\frac{2760}{\pi }\times \cos\left(\frac{\pi }{12}t\right)$$
- step5: Return the limits:
  $$\left(345t+\frac{2760}{\pi }\times \cos\left(\frac{\pi }{12}t\right)\right)\bigg |_{0}^{24}$$
- step6: Calculate the value:
  $$8280$$
The total energy consumed by the city in a typical 24-hour period is 8280 MWh.

To express the answer in Joules, we need to convert MWh to Joules. Given that 1 MWh is equal to $$ 3.6 \times 10^6 $$ Joules, we can convert 8280 MWh to Joules as follows:

$$ 8280 \, \text{MWh} = 8280 \times 3.6 \times 10^6 \, \text{J} $$

Now, we can calculate the total energy consumed in Joules.
Calculate the value by following steps:
- step0: Calculate:
  $$8280\times 3.6\times 10^{6}$$
- step1: Rewrite the expression:
  $$8280\times 36\times 10^{5}$$
- step2: Multiply the terms:
  $$298080\times 10^{5}$$
The total energy consumed by the city in a typical 24-hour period is $$ 2.9808 \times 10^{10} $$ Joules.

Therefore, the answers to parts (a) through (d) are:
a. The total energy consumed by the city in a typical 24-hour period is 8280 MWh.
b. The total energy consumed by the city in a typical 24-hour period is $$ 2.9808 \times 10^{10} $$ Joules.
The city consumes 8280 MWh of energy in a 24-hour period, which is equal to $$ 2.9808 \times 10^{10} $$ Joules.

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