5.4 Integration formulas and the net change theorem  (Page 6/8)

The following table lists the electrical power in gigawatts—the rate at which energy is consumed—used in a certain city for different hours of the day, in a typical 24-hour period, with hour 1 corresponding to midnight to 1 a.m.

Find the total amount of power in gigawatt-hours (gW-h) consumed by the city in a typical 24-hour period.

The average residential electrical power use (in hundreds of watts) per hour is given in the following table.

1. Compute the average total energy used in a day in kilowatt-hours (kWh).
2. If a ton of coal generates 1842 kWh, how long does it take for an average residence to burn a ton of coal?
3. Explain why the data might fit a plot of the form $p\left(t\right)=11.5-7.5\text{sin}\left(\frac{\pi t}{12}\right).$

The data in the following table are used to estimate the average power output produced by Peter Sagan for each of the last 18 sec of Stage 1 of the 2012 Tour de France .

Estimate the net energy used in kilojoules (kJ), noting that 1W = 1 J/s, and the average power output by Sagan during this time interval.

The data in the following table are used to estimate the average power output produced by Peter Sagan for each 15-min interval of Stage 1 of the 2012 Tour de France.

Estimate the net energy used in kilojoules, noting that 1W = 1 J/s.

The distribution of incomes as of 2012 in the United States in $5000 increments is given in the following table. The k th row denotes the percentage of households with incomes between $\mathrm{\$5000}xk$ and $5000xk+4999.$ The row $k=40$ contains all households with income between $200,000 and $250,000 and $k=41$ accounts for all households with income exceeding $250,000.

1. Estimate the percentage of U.S. households in 2012 with incomes less than $55,000.
2. What percentage of households had incomes exceeding $85,000?
3. Plot the data and try to fit its shape to that of a graph of the form $a\left(x+c\right)e^{\text{−}b\left(x+e\right)}$ for suitable $a,b,c.$

Newton’s law of gravity states that the gravitational force exerted by an object of mass M and one of mass m with centers that are separated by a distance r is $F=G\frac{mM}{r^2},$ with G an empirical constant $G=6.67x{10}^{\mathrm{-11}}{m}^3\text{/}\left(kg·s^2\right).$ The work done by a variable force over an interval $\left[a,b\right]$ is defined as $W={\displaystyle {\int }_a^{b}F\left(x\right)dx}.$ If Earth has mass $5.97219\times {10}^{24}$ and radius 6371 km, compute the amount of work to elevate a polar weather satellite of mass 1400 kg to its orbiting altitude of 850 km above Earth.