3.1.2: Exercise 3.1

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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) Units can be used to set up conversion problems by using the fact that common factors in the numerator and denominator of a fraction divide out to one.

(ii) It is important to know how many cents you are making per minute.

(iii) Units can be used to set up conversion problems by using the fact that common factors in the numerator and denominator of a fraction can be subtracted to equal zero.

(iv) It is possible to convert miles per hour to feet per second.

DEVELOPING SKILLS AND UNDERSTANDING

(2) Use Figure 1 for the following questions.

Figure 1

$3.1 figure 1.PNG$

(a) (i) What fraction of Figure 1 is shaded?

(ii) Shade the same fraction of the area in Figures 2 and 3.

Figure 2

$3.1 figure 2.PNG$

Figure 3

$3.1 figure 3.PNG$

(b) How many boxes did you shade for Figure 2?

(3) As of March 2023, Florence Griffith-Joyner held the women’s world record for the 100-meter dash. She set the record with a time of 10.49 seconds in 1988.⁴ Which of the following calculations are correctly set up to convert this speed into miles per hour?

(i) $\dfrac{10.49\;seconds}{100\;meters}\cdot\dfrac{1\;meter}{3.28\;feet}\cdot\dfrac{5,280\;feet}{1\;mile}\cdot\dfrac{1\;minute}{60\;seconds}\cdot\dfrac{1\;hour}{60\;minutes}$

(ii) $\dfrac{100\;meters}{10.49\;seconds}\cdot\dfrac{3.28\;feet}{1\;meter}\cdot\dfrac{1\;mile}{5,280\;feet}\cdot\dfrac{60\;seconds}{1\;minute}\cdot\dfrac{60\;minutes}{1\;hour}$

(iii) $\dfrac{10.49\;seconds}{10\;meters}\cdot\dfrac{1\;meter}{3.28\;feet}\cdot\dfrac{5,280\;feet}{1\;mile}\cdot\dfrac{60\;seconds}{1\;minute}\cdot\dfrac{60\;minutes}{1\;hour}$

(iv) $\dfrac{100\;meters}{10.49\;seconds}\cdot\dfrac{3.28\;feet}{1\;meter}\cdot\dfrac{1\;mile}{5,280\;feet}\cdot\dfrac{1\;minute}{60\;seconds}\cdot\dfrac{1\;hour}{60\;minutes}$

(4) Find the answer to the conversion in the previous question. Round to the nearest tenth of a mile per hour.

(5) A 2023 Toyota Prius hybrid vehicle gets 56 mpg for highway driving. The tank holds 11.3 gallons of fuel.⁵ Typically the low fuel warning light comes on when approximately two gallons of fuel remain in the tank. Which of the following calculations can be used to find the distance that can be traveled after the fuel light comes on and before the car runs out of gasoline?

(i) $\dfrac{1}{2\;gallons}\cdot\dfrac{56\;miles}{1\;gallon} = \large{28\;miles}$

(ii) $\dfrac{2\;gallons}{1}\cdot\dfrac{56\;miles}{1\;gallon} = \large{112\;miles}$

(iii) $\dfrac{11.3\;gallons}{1}\cdot\dfrac{56\;miles}{1\;gallon} = \large{632.8\;miles}$

(iv) $\dfrac{2\;gallons}{1}\cdot\dfrac{1\;gallon}{56\;miles} = \large{\frac{1}{28}miles}$

MAKING CONNECTIONS ACROSS THE COURSE

(6) In Collaboration 1.6, you learned about a water footprint. Part of a person’s water footprint is the water used for cleaning. In this question, you will calculate the cost of water for laundry and bathing. You will use the City of New York 2023 rate of $9.87/100 cubic feet of water. Calculate the cost of each of the following based on this rate. Use the conversion factor of 7.48 gallons per cubic foot.⁶

(a) A standard washing machine uses approximately 50 gallons of water per load.⁷ A household washes one load of laundry per week for 52 weeks. Find the total cost per year. Round to the nearest dollar per year.

(b) According to one study, the average American shower lasts for 8.2 minutes and uses 17.2 gallons of water.⁸ A person showers once a day for a year. Find the total cost per year. Round to the nearest dollar per year.

______________________________

⁴ https://olympics.com/en/news/evolution-womens-100m-world-record-flojo

⁵ https://www.toyota.com/prius/

⁶ https://www.nyc.gov/site/dep/water/residential-water-use.page

⁷ ibid.

⁸ ibid.

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