Accumulation with a Quadratic Function - Moore Public Schools

Connecting Algebra 2 to Advanced Placement* Mathematics 

A Resource and Strategy Guide 

Updated: 05/15/10 

Accumulation with a Quadratic Function 

Objective: 

Students will solve an application question involving accumulation of the areas of rectangles. 

Connections to Previous Learning: 

Students should know how to use a calculator to do quadratic regressions. 

Connections to AP*: 

AP Calculus Topic: Analysis of Functions; Accumulation 

AP Statistics Topic: Bivariate Data 

Materials: 

Student Activity pages, graphing calculators 

Teacher Notes: 

This lesson expands the concept of approximating the area of a bounded region using rectangles to 

include the notion that the approximation moves closer and closer to the actual value of the area 

when more and more rectangles of smaller and smaller widths are used. 

To complete this activity independently, students will need to have had experience with using 

rectangles to approximate area under curves and with using the calculator to do quadratic 

regressions. The summation notation introduced in this activity may be new for the students. Care 

should be taken to make sure that the students understand how this notation can be expanded to 

calculate the sum of the areas of the rectangles. 

*Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. 

The College Board was not involved in the production of this product. 

Copyright © 2008 Laying the Foundation ® , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 1


Student Activity 

Julián Guerrera, who owns a small farm near San Marcos, Texas, has recently installed a small tank 

on his property to provide flood control for the creek that flows near his home. Under normal 

circumstances no water flows into the tank; however, on a particularly rainy day, the creek rises and 

begins to flow into the tank. Mr. Guerrera collects data from his water gauge every hour beginning 

when the water first starts running into the tank at 8:00 a.m. He records the rate of flow in cubic 

meters per hour every hour for the first four hours of the storm. His table is provided below. The 

weather report predicts that the flood will continue to increase in its intensity until 1:00 p.m. and 

then quickly subside. Mr. Guerrera knows that if his tank, which can hold up to 22,000 cubic meters 

of water, is no more than half full at the end of the first five hours, his home is safe from the flooding 

water. Eduardo, Mr. Guerrera’s son, is taking Algebra II, so he volunteers to monitor the gauge and 

to evaluate the data. He, of course, needs your help. The gauge readings taken from 8:00 a.m. to 

noon are recorded in the table provided below. 

8:00 a.m. 9:00 a.m. 10:00 a.m. 11:00 a.m. 12:00 noon 

500 m 3 

/hr. 

3 

700 m /hr. 

3 

1300 m /hr. 

2300 m /hr. 

Copyright © 2008 Laying the Foundation ® , Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 2 

3 

3 

3700 m /hr. 

1. The first task is to create a graph of the data. 

a) Code the data so that 8:00 a.m. is at time 0 and so that 500 m 3 

/hr is represented by 5. 

b) Sketch a scatterplot of the data. 

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20 

15 

10 

5 

0 

0 

1 2 3 4 5 6 

c) Label the axes with words.


d) Draw a smooth continuous curve to connect the data points for this time period. Explain 

how the statement about the weather report given in the opening paragraph indicates that 

a smooth continuous curve may be drawn for the given time period. 

e) What type of function does the data appear to represent? 

2. Since the flood is predicted to subside after 5 hours, you need to know the rate at which the 

water will be flowing at 1:00 p.m.; however, you cannot wait until then to start moving 

furniture. You decide to take a chance on the fact that a quadratic function will fit the data 

relatively well. 

a) Enter the data into L1 and L2 on your graphing calculator and adjust the window to 

display the data. 

b) Calculate a quadratic regression on the data, and graph the function to make sure that it 

gives an acceptable fit. 

c) What is the function? 

d) Using your function, calculate the rate at which the water is flowing into the tank at 

1:00 p.m. 

e) Add this data point to your table and to your graph. 

3. The information that has been gathered represents the rate of flow of the water in cubic meters 

per hour, but you need to calculate the total amount of water that will collect in the tank over 

the five hour period. You make a quick phone call to your teacher, and she suggests that what 

you should approximate the area under the curve. 

a) Consider the units of the x-axis (time-axis) and the units of the y-axis (rate-axis). If you 

calculate the area of a rectangle on this grid, what units will be on the area? 

b) Following your teacher’s suggestion, draw 5 vertical line segments, one at each hour, 

from the time-axis to the curve. Now draw five horizontal segments beginning on the 

left-side of the curve to form 5 rectangles so that the height of the first rectangle is 5, the 

second is 7, the third is 13, etc. (Since this is a continually increasing function, the 

rectangles will all lie completely below the curve.) 

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10 

5 

0 

0 

1 2 3 4 5 6 


c) What is the area (include the units) of the first rectangle? ________ 

d) What is the area (include the units) of the second rectangle? ________ 

e) What is the area (include the units) of the third rectangle? ________ 

f) What is the area (include the units) of the fourth rectangle? ________ 

g) What is the area (include the units) of the fifth rectangle? ________ 


h) Total the areas of the five rectangles, and explain what this measurement represents. 

i) Is this measurement too small or too large? Explain your choice based on the graph that 

you drew. 

4. a) Draw another sketch of the data; however, this time sketch in rectangles that use the 

right-hand side of the region for the height. These are called “right-hand” rectangles. 

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5 

0 

0 

1 2 3 4 5 6 

b) What is the area (include the units) of the first rectangle? ________ 

c) What is the area (include the units) of the second rectangle? ________ 

d) What is the area (include the units) of the third rectangle? ________ 

e) What is the area (include the units) of the fourth rectangle? ________ 

f) What is the area (include the units) of the fifth rectangle? ________ 

g) Total the areas of the five rectangles, and explain what this measurement represents. 

h) Is this approximation too small or too large? Explain your choice based on the graph that 

you drew. 


5. According to your calculation the amount of water in the tank will be between 

_________________________ and ________________________. 


If the amount of water is greater than ____________ , the Guerrera family needs to prepare 

the house for flooding so the situation looks bad. Your under estimation looks acceptable; 

however, the over estimation means that you need to help move furniture to the second floor 

of the house. 

6. You and Eduardo decide to refine your estimate instead of moving furniture; you need more 

data points in order to get rectangles with smaller widths. 

a) You make a new table and add in values for every half-hour by calculating the rate 

predicted by your function at 0.5, 1.5, 2.5, 3.5, and 4.5. 

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 

5 7 13 23 37 55 

b) You sketch a new graph and draw in 10 right-hand rectangles to represent a new upper 

bound for the amount of water in the tank. 

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5 

0 

0 

1 2 3 4 5 6 

c) Use the 10 right-hand rectangles to calculate the new upper bound for the amount of 

water in the tank. 

This is better, but it still predicts that you must move furniture. 



7. To simplify your work, you decide to use the mathematical notation for this process which is 

last t-value 

∑ 

first t-value, (step= ∆t) 

f() t ⋅∆t 

. 

This says to sum the product of the height and the width of the rectangles beginning at t = first 

t-value and stopping at t = last t-value, making a calculation every ∆t units. 

5−0 ∆ t = , where n is the number of rectangles that you want to use. 

n 

To use this notation for the set of 5 left-hand rectangles you write 

4 

∑ 

t= 

0, (step = 1) 

2 

(2t + 5) ⋅1 

(Notice that 0 is the first value of t for the left-hand rectangles) 

To use this notation for the set of 5 right-hand rectangles you write 

5 

∑ 

t= 

1, (step = 1) 

2 

(2t + 5) ⋅1 

(Notice that 1 is the first value of t for this set of right-hand rectangles) 

The set of 10 right-hand rectangles is written as 

5 

∑ 

t= 

0.5, (step = 0.5) 

t 

+ ⋅ = 

2 

(2 5) 0.5 

a) Explain why 0.5 is the first t-value for this set of right-hand rectangles. 

b) Complete the expanded version of this statement 

2 2 2 

(2(0.5) + 5) ⋅ 0.5 + (2(1) + 5) ⋅ 0.5 + (2(1.5) + 5) ⋅ 0.5 + 



8. You remember that there is a way to do this with your calculator, but you cannot remember 

the details. Again you call your teacher for assistance, and she reminds you that the calculator 

has a set of commands that will allow you to sum a sequence. She tells you to make sure that 

your equation is entered in y1 of your calculator and then to go back to the home screen. 

On the calculator 

last x 

∑ f( x) ⋅∆x 

is entered as sum(seq (f ( x) × ∆x, x, first x, last x, ∆ x) 

) . 

first x,(step = ∆x) 

For right-hand rectangles 

5 

∑ y1⋅∆t is entered as sum(seq (y1× ∆xx , , 0+ ∆x, 5, ∆ x)) 

. 

0 +∆t,(step = ∆t) 

You decide to check that you understand the process by repeating your calculation of 10 

5−0 right-hand rectangles. ∆ t = = .5, 

and the first t-value is 0.5. 

10 

Remember that in your calculator the equation is written as a function of x instead of t. 

On a TI83 or a TI84 calculator: 

Press 2 nd 

Stat List 

Arrow over to Math 

Choose #5 Sum( 

Press 2 nd 

Stat List again 

Arrow over to OPS 

Choose #5 Seq( Your screen now shows sum(seq( 

Press VARS 

Arrow over to Y-VARS 

Choose #1 Function 

Select #1 Y1 Your screen now shows sum(seq(y1 

Type × .5, 

comma 

Complete the statement by typing 

x, comma, 0.5, comma, 5, comma, and 0.5, close parenthesis. 

Your screen now shows sum(seq (y1×0.5, x , 0.5, 5, 0.5)) 

Press enter to get the total area with 10 rectangles. 

Remember that you have coded your data so that you will need to multiply the final answer 

by 100 to get back to the original units. 

Check to be sure that your answers agree. 


9. You decide to try using 20 rectangles. 

a) What is the value of ∆ t for 20 rectangles? 

b) What is the first t-value for 20 right-hand rectangles? 

c) What is the mathematical notation for summing 20 right-hand rectangles? 

d) Using your calculator, calculate the amount of water predicted by 20 right-hand 

rectangles. Do you still have to move furniture? 


10. Continue the process, doubling the number of rectangles, until you can show that the amount 

of water in the tank at 1:00 p.m. will be less than 11,000 cubic meters. 

a) Your final predicted amount of water is ____________________ . 

b) The final number of rectangles used is ___________ . 

c) The final width of each rectangle is ____________. 

11. Write a summary of the results of the activity that explains the thinking process used to get 

the final results. 


Connecting Algebra II to Advanced Placement* Mathematics 

A Resource and Strategy Guide 


Answers: 

1. a) 0 1 2 3 4 

5 7 13 23 37 

b), c), d) 

2. a) 

100 cubic meters per hour 

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15 

10 

5 

0 

1 2 3 4 5 6 

Hours after 8:00 am 

e) Use the table and calculate the difference between each of the y-values. (2, 6, 10, 14) 

b) 

c) 

0 1 2 3 4 

5 7 13 23 37 

Calculate the second difference between the y values which is always 4. Since the 

second difference is the same, the function is quadratic. 

y = x + 

2 

2 5 

d) At 1:00 p.m. there the water will be increasing at 5500 m 3 

/hour. 

e) 


3. a) The units will be 100 

4. a) 


3 

c) 500 m 

3 

d) 700 m 

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5 

3 

e) 1300 m 

3 

f) 2300 m 

3 

g) 3700 m 


0 

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15 

10 

5 

0 

3 

meters 3 

hour = meters 

hour 

1 2 3 4 5 6 


h) The total area represents 8500 m 3 

. 

1 2 3 4 5 6 


⋅ . 

i) This measurement is too small; the rectangles all fall below the curve leaving space 

between the curve and the rectangles. 

j) The data is coded so that the scale is 100 cubic meters. 

Answers 


5. 

6. a) 

3 

b) 700 m 

3 

c) 1300 m 

3 

d) 2300 m 

3 

e) 3700 m 

3 

f) 5500 m 

3 

g) The total area represents 13,500 m 

h) This measurement is too large; the rectangles include space that is above the curve. 

Answers 

3 3 

8,500 m ≤ amount of water ≤ 13,500 m If the amount of water is greater than 11,000 m 3 

at 1:00 p.m. the house will probably be flooded. 

b) The drawing is similar to the one in 4a with widths of 0.5 instead of 1. 

3 

c) 12,125 m 

7. a) Since the rectangles are right-hand rectangles, the height is calculated on the right side of 

the rectangle thus 0.5 is the right side of the first rectangle. 

b) 

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 

5 5.5 7 9.5 13 17.5 23 29.5 37 45.5 55 

2 

(2(0.5) 5) 0.5 

2 

(2(1) 5) 0.5 

2 

(2(1.5) 5) 0.5 

2 2 2 

(2(2.0) + 5) ⋅ 0.5 + (2(2.5) + 5) ⋅ 0.5 + (2(3.0) + 5) ⋅ 0.5 + 

2 

(2(3.5) 5) 0.5 

2 

(2(4.0) 5) 0.5 

2 

(2(4.5) 5) 0.5 

2 

(2(5.0) 5) 0.5 

2.75+3.5+4.75+6.5+8.75+11.5+14.75+18.5+22.75+27.5 = 121.25 or 12,125 m 3 

. 

3 

8. sum(seq(y1*0.5, x, 0.5, 5, 0.5)) = 121.25 or 12,125 m 

5−0 9. a) ∆ t = = .25 

20 

b) 0.25 

+ ⋅ + + ⋅ + + ⋅ + 

+ ⋅ + + ⋅ + + ⋅ + + ⋅ = 


5 

2 

c) ∑ ( t ) 

t= 

0.25, (step = 0.25) 

2 + 5 × 0.25 

Answers 

d) 11,468.75 m 3 . This amount is still more than 11,000 m 3 

; therefore, the furniture still has 

to be moved. 

10. a) The sum is 10,990 m 

b) 80 rectangles 

c) Width 0.0625 hours. 

3 

11. Mr. Guerrera explained that if the water had not filled more than half of the tank by 1:00 p.m., 

the predicted peak of the flood, that the tank would hold the flood water. In order to calculate 

the amount of water that would be collected by 1:00 p.m. Eduardo and I used the flood gauge 

data that gave the rate of flow for the water in m 3 /hour. Since we had four data points, we 

plotted the curve and predicted that a quadratic function would be a good fit for the data. We 

used the graphing calculator to do a quadratic regression on the data, then, with this equation, 

we predicted the flow rate at 1:00 p.m. to be 5500 m 3 /hour. Using 80 right-hand rectangles of 

width 0.0625 hours, we calculated an upper bound for the amount of water that would be in 

the tank to be 10,990 m 3 . Since this amount is less than the critical amount of 11,000 m 3 

, we 

predicted that the tank would safely hold the flood water.

Accumulation with a Quadratic Function - Moore Public Schools

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