Question

problems 8 - 9: here are two recipes for making sparkling lemonade. for each recipe, determine how many tablespoons of lemonade mix are required per cup of sparkling water. show or explain your thinking.

1. 4 tablespoons of lemonade mix and 12 cups of sparkling water
2. 4 tablespoons of lemonade mix and 6 cups of sparkling water
3. select all of the coordinate pairs that could be part of a proportional relationship with a constant of proportionality of 4.

□ a. (3, 12)
□ b. (4, 8)
□ c. (3.5, 14)
□ d. (6, 10)
□ e. (\\(\frac{1}{2}\\), 2)

1. here is triangle def and its dimensions. which measurements represent the dimensions of a triangle that is a scaled copy of triangle def?

a. 20 cm, 12 cm, 24 cm
b. 9 cm, 5 cm, 10 cm
c. 27 cm, 16 cm, 33 cm
d. 13.5 cm, 7.5 cm, 16.5 cm

Explanation:

Step1: Analyze problem 8

To find the number of tablespoons of lemonade mix per cup of sparkling water, divide the number of tablespoons of lemonade mix by the number of cups of sparkling water.
For problem 8: \( \frac{4}{12}=\frac{1}{3}\) (tablespoons per cup)

Step2: Analyze problem 9

For problem 9: \( \frac{4}{6}=\frac{2}{3}\) (tablespoons per cup)

Step3: Analyze problem 10

In a proportional relationship \(y = kx\) (where \(k\) is the constant of proportionality).

• For option A: If \(x = 3\), \(y=kx\), \(k = 4\), then \(y=4\times3 = 12\). So \((3,12)\) is valid.
• For option B: If \(x = 4\), \(y=4\times4=16

eq8\). So \((4,8)\) is not valid.

• For option C: If \(x = 3.5\), \(y=4\times3.5 = 14\). So \((3.5,14)\) is valid.
• For option D: If \(x = 6\), \(y=4\times6=24

eq10\). So \((6,10)\) is not valid.

• For option E: If \(x=\frac{1}{2}\), \(y = 4\times\frac{1}{2}=2\). So \((\frac{1}{2},2)\) is valid.

Step4: Analyze problem 11

For a scaled - copy, the ratios of corresponding sides should be equal.
The sides of \(\triangle DEF\) are \(10\), \(18\), \(22\).

• For option D:

\(\frac{13.5}{18}=\frac{13.5\div4.5}{18\div4.5}=\frac{3}{4}\), \(\frac{7.5}{10}=\frac{7.5\div2.5}{10\div2.5}=\frac{3}{4}\), \(\frac{16.5}{22}=\frac{16.5\div5.5}{22\div5.5}=\frac{3}{4}\)

Answer:

1. \(\frac{1}{3}\) tablespoon per cup.
2. \(\frac{2}{3}\) tablespoon per cup.
3. A. \((3,12)\), C. \((3.5,14)\), E. \((\frac{1}{2},2)\)
4. D. \(13.5\mathrm{cm},7.5\mathrm{cm},16.5\mathrm{cm}\)