Question

write an equation to describe the relationship between the distance she runs in miles, d, and her running speed, in miles per hour, when she runs:
a. at a constant speed of 4 miles per hour for the entire 30 minutes.

b. at a constant speed of 5 miles per hour the first 20 minutes, and then at 4 miles per hour the last 10 minutes.

c. at a constant speed of 6 miles per hour the first 15 minutes, and then at 5.5 miles per hour for the remaining 15 minutes.

d. at a constant speed of a miles per hour the first six minutes, and then at 6.5 miles per hour for the remaining 24 minutes.

e. at a constant speed of 5.4 miles per hour for m minutes, and then at b miles per hour for n minutes.

8 from unit 4, lesson 3
in the 21st century, people measure length in feet and meters. at various points in history, people measured length in hands, cubits, and paces. there are 9 hands in 2 cubits. there are 5 cubits in 3 paces.

a. write an equation to express the relationship between hands, h, and cubits, c.

Explanation:

Step1: Convert time to hours

30 minutes = $\frac{30}{60} = 0.5$ hours

Step2: Apply distance formula ($D=speed\times time$)

$D = 4\times0.5$
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Step1: Convert times to hours

20 minutes = $\frac{20}{60} = \frac{1}{3}$ hours; 10 minutes = $\frac{10}{60} = \frac{1}{6}$ hours

Step2: Calculate total distance

$D = 5\times\frac{1}{3} + 4\times\frac{1}{6}$
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Step1: Convert times to hours

15 minutes = $\frac{15}{60} = 0.25$ hours

Step2: Calculate total distance

$D = 6\times0.25 + 5.5\times0.25$
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Step1: Convert times to hours

6 minutes = $\frac{6}{60} = 0.1$ hours; 24 minutes = $\frac{24}{60} = 0.4$ hours

Step2: Calculate total distance

$D = a\times0.1 + 6.5\times0.4$
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Step1: Convert times to hours

$m$ minutes = $\frac{m}{60}$ hours; $n$ minutes = $\frac{n}{60}$ hours

Step2: Calculate total distance

$D = 5.4\times\frac{m}{60} + b\times\frac{n}{60}$
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Step1: Set up proportional relationship

Given 9 hands = 2 cubits, so $\frac{h}{c} = \frac{9}{2}$

Step2: Rearrange to solve for $h$

$h = \frac{9}{2}c$

Answer:

a. $D = 2$
b. $D = \frac{5}{3} + \frac{2}{3} = 2$
c. $D = 1.5 + 1.375 = 2.875$
d. $D = 0.1a + 2.6$
e. $D = \frac{5.4m}{60} + \frac{bn}{60}$
8a. $h = \frac{9}{2}c$ (or $2h = 9c$)