Math

Question Find the weight-to-volume ratio of water. Given that 50 gallons of water weigh 417 pounds, express the relationship between the volume vv (in gallons) and weight ww (in pounds) as an equation.

Studdy Solution

STEP 1

Assumptions
1. The weight of water (ww) is proportional to its volume (vv).
2. 50 gallons of water weigh 417 pounds.
3. We need to find the unit rate, which is the ratio of weight to volume.

STEP 2

Since the weight of water is proportional to its volume, we can express this relationship as:
w=kvw = k \cdot v
where kk is the constant of proportionality.

STEP 3

To find the constant of proportionality kk, we use the given information that 50 gallons of water weigh 417 pounds.
417 pounds=k50 gallons417 \text{ pounds} = k \cdot 50 \text{ gallons}

STEP 4

Solve for the constant of proportionality kk by dividing both sides of the equation by 50 gallons.
k=417 pounds50 gallonsk = \frac{417 \text{ pounds}}{50 \text{ gallons}}

STEP 5

Calculate the constant of proportionality kk.
k=41750 pounds per gallonk = \frac{417}{50} \text{ pounds per gallon}

STEP 6

Simplify the fraction to find the unit rate in pounds per gallon.
k=8.34 pounds per gallonk = 8.34 \text{ pounds per gallon}

STEP 7

Now that we have the constant of proportionality, we can write the equation that relates the volume of water vv and the weight of the water ww.
w=8.34vw = 8.34 \cdot v

STEP 8

Express the equation in terms of a fraction and then as a decimal.
As a fraction: w=41750vw = \frac{417}{50} \cdot v
As a decimal: w=8.34vw = 8.34 \cdot v
Include proper units: w (pounds)=8.34 (pounds per gallon)v (gallons)w \text{ (pounds)} = 8.34 \text{ (pounds per gallon)} \cdot v \text{ (gallons)}
(a) The ratio of the weight of water to its volume as a unit rate is 8.348.34 pounds per gallon. (b) The equation that relates vv and ww is w=8.34vw = 8.34 \cdot v.

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