Question

15.2 comparing baths and showers
some people say that it uses more water to take a bath than a shower. others disagree.

1. what information would you need in order to answer the question?
2. describe how you could get the information and how you would use the information to find the answer.
3. find out values for the measurements you need to use the method you described. you may ask your teacher or research them yourself.
4. under what conditions does a bath use more water? under what conditions does a shower use more water? explain or show your reasoning.

15.3 representing water usage

1. continue considering the problem from the previous activity. name two quantities that are in a proportional relationship. explain how you know they are in a proportional relationship.
2. what are two constants of proportionality for the proportional relationship? what do they tell us about the situation?
3. on graph paper, create a graph that shows how the two quantities are related. make sure to label the axes.
4. write two equations that relate the quantities in your graph. make sure to record what each variable represents.

Explanation:

15.2 Comparing Baths and Showers

Step1: Identify needed information

We need the volume of water used in a bath (volume of bathtub filled) and the volume of water used in a shower (flow - rate of showerhead and duration of shower).

Step2: Information - gathering method

To get the volume of water for a bath, measure the dimensions of the bathtub and calculate its volume using the formula for the volume of a rectangular - prism (if bathtub is rectangular - like) $V = l\times w\times h$. For a shower, measure the flow - rate of the showerhead (e.g., in gallons per minute) and time the duration of a typical shower.

Step3: Find measurement values

Typical bathtub dimensions: length $l = 5$ feet, width $w = 2.5$ feet, height $h = 1.5$ feet. Volume of bathtub $V_{bath}=l\times w\times h=5\times2.5\times1.5 = 18.75$ cubic feet. 1 cubic foot is approximately 7.48 gallons, so $V_{bath}\approx18.75\times7.48 = 140.25$ gallons. A typical showerhead flow - rate is 2.5 gallons per minute, and a shower might last 10 minutes, so $V_{shower}=2.5\times10 = 25$ gallons.

Step4: Analyze water - usage conditions

A bath uses more water when the volume of water needed to fill the bathtub is greater than the volume of water used during a shower. This depends on the size of the bathtub, the flow - rate of the showerhead, and the duration of the shower. If the shower is very long or the bathtub is very small, a shower could use more water.

15.3 Representing Water Usage

Step1: Identify proportional quantities

Two quantities that could be in a proportional relationship are the duration of the shower and the volume of water used during the shower (assuming a constant flow - rate). Another could be the number of baths and the total volume of water used for those baths (assuming the same bathtub is used and filled to the same level each time).

Step2: Determine constants of proportionality

For the shower, if $V$ is the volume of water and $t$ is the time of the shower, and the flow - rate $r$ is constant, the constant of proportionality is the flow - rate of the showerhead. For example, if $r = 2.5$ gallons per minute, then $V=rt$. For baths, if $V_{total}$ is the total volume of water for $n$ baths and $V_{single}$ is the volume of water for one bath, the constant of proportionality is $V_{single}$. It tells us how much water is used per unit (per minute for shower, per bath for baths).

Step3: Create a graph

If we consider the shower example with volume of water $V$ on the y - axis and time $t$ on the x - axis, and a flow - rate of 2.5 gallons per minute. The graph is a straight line passing through the origin (since when $t = 0$, $V = 0$). We can label the x - axis as "Time of Shower (minutes)" and the y - axis as "Volume of Water Used (gallons)".

Step4: Write equations

For the shower example, if $V$ is the volume of water and $t$ is the time of the shower with a flow - rate $r = 2.5$ gallons per minute, the equations are $V = 2.5t$ and $t=\frac{V}{2.5}$.

Answer:

For 15.2:

• Information needed: Volume of bath water and volume of shower water.
• Method: Measure bathtub dimensions and shower - head flow - rate and duration.
• Values example: Bath $\approx140.25$ gallons, Shower = 25 gallons.
• Conditions: Depends on bathtub size, shower - head flow - rate, and shower duration.

For 15.3:

• Proportional quantities: Shower time and volume of shower water; number of baths and total bath water volume.
• Constants of proportionality: Shower flow - rate; volume of one bath.
• Graph: Straight - line graph with appropriate axis labels for shower example.
• Equations: For shower $V = 2.5t$ and $t=\frac{V}{2.5}$ (assuming $r = 2.5$ gallons per minute).