3.8.2: Exercise 3.8

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MAKING CONNECTIONS TO THE COLLABORATION

(1) Which of the following was one of the main mathematical ideas of the collaboration?

(i) The order of operations is used in determining the order of steps in solving an equation.

(ii) The order of operations is not related to solving equations.

(iii) Weight, gender, and time are all important factors in Blood Alcohol Content (BAC).

(iv) You can undo addition by subtracting.

DEVELOPING SKILLS AND UNDERSTANDING

(2) Solve the following equations for the unknown variable:

(a) 12.4 = 3w + 8 – 0.5w

(b) 3(r + 1.6) – 8r = 9.5

(d) 3 = $\dfrac{18 + n}{44}$

(3) Solve for the specified variable in each equation.

(a) A group of French students plan to visit the United States for two weeks. They are trying to pack appropriate clothing, but are not familiar with Fahrenheit. One student remembers this formula:

\[F = \dfrac{9}{5}C + 32\nonumber \]

where F is the temperature in Fahrenheit and C is the temperature in Celsius. Solve the equation for C.

(b) Recall using the simple interest formula, A = P + Prt, from Exercise 3.4. In the formula:

A = the full amount paid for the loan

P = the principal or the amount borrowed

r = the interest rate as a decimal

t = time in years

A car dealership wants to use the formula to find the rate needed for certain values of the other variables. Solve the formula for r.

(4) Akiko earns $2335 per month at her full time job. She also works part-time on weekends and evenings for $10.70/hour.

(a) Write an equation for Akiko’s monthly income, M. Use h for the number of hours she works part-time.

(b) Akiko would like to set up a spreadsheet that will calculate how many hours she has to work on weekends and evenings to earn different amounts. Her spreadsheet is shown below.

Screenshot of spreadsheet with two columns, A and B. Cell A1 = Monthly income. Cell B1 = Hours. A2 = 2400. A3 = 2500. A4 = 2600. A5 = 2700. A6 = 2800. B cells are blank.

Write a formula that Akiko can use in cell B2 to calculate the hours. Test your formula on a
calculator or spreadsheet to make sure it is correct.

(c) Assume there are 4 weeks in a month. Assume that she will be working approximately the same number of hours each week. How many hours does Akiko need to work on weekends and evenings each week to earn $2,600 per month?

(5) Recall the simplified formula for the braking distance of a car:

\[d = 0.018268 \cdot V^{2}_{0}\nonumber \]

where V0 is the initial velocity of the car (feet per second) and d is the braking distance (ft). In this model, the roadway grade is kept constant at 5% and the coefficient of friction at 0.8. In a school zone, you want the maximum braking distance to be 10 feet since this seems like a reasonable distance to see a child who might be in the way of a driver.

(a) What should you set as the speed limit (i.e., initial velocity of the car) so that the braking distance is 10 feet or less? Round to the nearest foot per second.

(b) The equation x² = 9 has two solutions: 3 and -3.

(i) With that in mind, what is the other solution to the equation 10 = 0.018268 • $V^{2}_{0}$, besides what you answered in (a)?

(ii) Is this a valid solution in the context of this problem?

MAKING CONNECTIONS ACROSS THE COURSE

(6) Refer to Collaboration 1.6 in which you compared the water footprint of different countries. The following information was given. Use the table to answer the questions below.

Country	Population (in thousands)	Total Water Footprint²⁵ (in 10⁹ cubic meters per year)
China	1,277,208	1,368
India	1,051,290	1,145

(a) What does 10⁹ cubic meters mean?

(i) One trillion cubic meters

(ii) One billion cubic meters

(iii) One million cubic meters

(iv) One hundred thousand cubic meters

(b) The following equation is based on information from the table.

$\dfrac{1,368}{1,277,208} = \dfrac{x}{1,051,290}$

What does x represent?

(i) The water footprint of China if it used water at the same rate as India.

(ii) The population of China if it used water at the same rate as India.

(iii) The water footprint of India if it used water at the same rate as China.

(iv) The population of India if it used water at the same rate as China.

__________________________________________

²⁵ http://www.waterfootprint.org/Reports/Report50-NationalWaterFootprints-Vol1.pdf

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