Problem 60 - College Algebra with Modeling and Visualization

Question

Rainfall Suppose that during a storm rain is falling at a rate of 1 inch per hour. The water coming from a circular roof with a radius of 20 feet is running down a downspout that can accommodate 400 gallons of water per hour. See the figure. (a) Determine the number of cubic inches of water falling on the roof in 1 hour. (b) One gallon equals about 231 cubic inches. Write a formula for a function \(g\) that computes the gallons of water landing on the roof in \(x\) hours. (c) How many gallons of water land on the roof during a 2.5 -hour rain storm? (d) Will one downspout be sufficient to handle this type of rainfall? How many downspouts should there be? (IMAGE CANT COPY)

Step-by-Step Solution

Answer

(a) 57600π cubic inches. (b) \( g(x) = x \times \frac{57600\pi}{231} \). (c) Use \( g(2.5) \). (d) Compare \( g(2.5) \) to 1000 gallons.

Step 1: Calculate Area of the Roof

To find the area of a circular roof, use the formula for the area of a circle: \( A = \pi r^2 \). With a radius \( r = 20 \) feet, the area in square feet is \( A = \pi \times (20)^2 = 400\pi \) square feet.

Step 2: Convert Area to Square Inches

Since we are aiming to find cubic inches and 1 foot = 12 inches, we convert the area to square inches. \( 400\pi \text{ square feet} \times (12^2) \text{ square inches per square foot} = 57600\pi \) square inches.

Step 3: Calculate Cubic Inches of Rainfall

The volume of water falling in 1 hour is the area in square inches times the rainfall rate (1 inch per hour): \( V = 57600\pi \times 1 = 57600\pi \) cubic inches.

Step 4: Define Function for Water in Gallons

A gallon is 231 cubic inches. Define function \( g(x) \) for the gallons falling in \( x \) hours. First find gallons per hour: \( \text{Gallons per hour} = \frac{57600\pi}{231} \). Then, \( g(x) = x \times \frac{57600\pi}{231} \).

Step 5: Find Gallons for 2.5 Hours

Using \( g(x) \) function, compute gallons for 2.5 hours: \( g(2.5) = 2.5 \times \frac{57600\pi}{231} \). Calculate the result.

Step 6: Determine Downspout Requirement

Since the downspout can handle 400 gallons per hour, find the total water from the storm by computing \( g(2.5) \) as in Step 5. Divide total water by 400 to find the number of downspouts needed.

Key Concepts

Cubic Inches Calculation

When examining rainfall problems, a vital part is understanding how to calculate the volume of water falling on an area using cubic inches. This starts by determining the area that the rainfall is impacting. Once that area is known, you can calculate the volume of rainfall over time. In this scenario, we have a circular roof where the rain falls.
To calculate the volume, first compute the area of the circular roof in square inches, which involves a conversion since measurements might originally be in feet. Then, use the rainfall rate (typically inches per hour in these problems) to determine the cubic inches.
For example, given a rainfall rate of 1 inch per hour, the volume of rain is essentially how thick the layer of water would be over the entire area. Multiplying this thickness by the area gives the cubic inches — that is, the volume — of rain. This fundamental understanding is key to solve other parts of the problem efficiently.

Circular Roof Area

The circular roof area plays an essential role in solving rainfall problems. Calculating the area of a circular shape involves using the formula for the area of a circle: \[ A = \pi r^2 \]where \( A \) represents the area and \( r \) is the radius. For this exercise, the radius is given in feet.
Once you've calculated the area in square feet, it's often necessary to convert this to square inches when dealing with real-world scenarios like rain volume. Why square inches? It's because we need to express the volume of rain in cubic inches.
Remember the conversion factor: 1 foot = 12 inches, meaning you must square this conversion factor to account for square feet to square inches. So, by multiplying your area in square feet by \( 12^2 \), you convert to square inches, ready for the next steps in volume calculation.

Water Flow Rate

Understanding the water flow rate is crucial when assessing how much water is collected or needs to be handled by drainage systems. Water flow rate, commonly measured in gallons per hour, indicates the volume of water a system like a downspout can handle.
In the problem's context, you first determine the total rainfall in cubic inches over a set time period — say 1 hour. Then, you convert this volume to gallons knowing that 1 gallon equals 231 cubic inches.
Once you've established the flow rate in gallons per hour, you can easily scale this up by adjusting for longer rainfall periods. This helps determine if the current drainage system is sufficient or if more capacity is needed.

Downspout Capacity

Assessing downspout capacity involves comparing the water output from your surface area with what the drainage hardware can manage. Downspouts have a maximum capacity, typically measured in gallons per hour.
With the given water flow rate, you need to calculate whether a single downspout is adequate. Start by finding out how many gallons are produced by the roof over the entire rain event duration.
Using previously calculated figures, divide the total gallons of water by the downspout's capacity to see how many downspouts are necessary. If the number exceeds one, additional downspouts will be needed to prevent overflow and potential water damage. This calculation ensures that rainwater is properly managed, avoiding water accumulation problems.