The cooking club sells stickers featuring Featherbrain the Turkey, the school's mascot. In one hour, they sell $fifty five worth of stickers. Each member who sells stickers is paid $two. Represent earnings from selling stickers as positive and costs of paying members as negative.
Define a unit for the amount of money the club will make.
How much money will the club make if nine members sell stickers?
How much money will the club make if six members sell stickers?
Complete the rows for the amount paid to each member and the income from selling stickers. Then, enter a variable for the number of members who sell stickers and use this variable to write an expression for the amount of money the club will make.
Use the Worksheet to complete the problem. The column labels describe the quantity from the problem scenario that the column is about. The row labels describe the type of value that goes in the row: quantity descriptions, units of measure, algebraic expressions, or numeric values to answer a question.
Quantity Name
Members
Money Made
Unit
members
How much money will the club make if nine members sell stickers?
Question 1
How much money will the club make if six members sell stickers?
Question 2
Cost of paying each member
dollars per member
Earnings from selling stickers
dollars
Expression
Quantity Name | Members | Money Made
--------------------------------------------
Unit | members | $
--------------------------------------------
How much money will the club make if nine members sell stickers? | 9 | $
--------------------------------------------------------------
How much money will the club make if six members sell stickers? | 6 | $
--------------------------------------------------------------
Cost of paying each member | | -$2/member
--------------------------------------------------------------
Earnings from selling stickers | | +$55
--------------------------------------------------------------
Expression | x | $55 - $2x
How much money will the club make if nine members sell stickers?
How much money will the club make if six members sell stickers?
To find out how much money the club will make if nine members sell stickers, we can substitute 9 for x in the expression:
Money made = $55 - $2x
Money made = $55 - $2(9)
Money made = $55 - $18
Money made = $37
Therefore, the club will make $37 if nine members sell stickers.
To find out how much money the club will make if six members sell stickers, we can substitute 6 for x in the expression:
Money made = $55 - $2x
Money made = $55 - $2(6)
Money made = $55 - $12
Money made = $43
Therefore, the club will make $43 if six members sell stickers.
After her alarm rings, Daisy has fifty-five minutes to get ready. She stays in bed one minute after the alarm rings the first day. She stays in bed for two minutes after the alarm rings the second day. She stays in bed three minutes after the alarm rings the third day, and so on.
Define units for the number of days and the time Daisy has to get ready.
How much time will she have to get ready after the alarm rings on the 29th day?
When Daisy is down to sixteen minutes to get ready, how many days has it been?
Complete the rows for the number of additional minutes that Daisy snoozes each day and the time Daisy has to get ready when her alarm rings. Then, enter a variable for the number of days and use this variable to write an expression for the time Daisy has to get ready.
Use the Worksheet to complete the problem. The column labels describe the quantity from the problem scenario that the column is about. The row labels describe the type of value that goes in the row: quantity descriptions, units of measure, algebraic expressions, or numeric values to answer a question.
Quantity Name
Number of Days
Time to Get Ready
Unit
How much time will she have to get ready after the alarm rings on the 29th day?
Question 1
When Daisy is down to sixteen minutes to get ready, how many days has it been?
Question 2
Change in time to get ready each day
minutes per day
Time to get ready when alarm rings
minutes
Expression
Quantity Name | Number of Days | Time to Get Ready
-----------------------------------------------------------------
Unit | days | min
-----------------------------------------------------------------
How much time will she have to get ready after the alarm rings on the 29th day? | 29 | $
---------------------------------------------------------------------------------------
When Daisy is down to sixteen minutes to get ready, how many days has it been? | | 16
---------------------------------------------------------------------------------------
Change in time to get ready each day | | +1 minute/day
---------------------------------------------------------------------------------------
Time to get ready when alarm rings | x | $
---------------------------------------------------------------------------------------
Expression | x | 55 - (1+2+3+...+(x-1))
Note: The formula for the sum of the first n positive integers is n(n+1)/2. Therefore, the sum of the first (x-1) positive integers is (x-1)(x-1+1)/2, which simplifies to (x-1)x/2 or (x^2-x)/2.
So, the expression for the time Daisy has to get ready when the alarm rings is:
Time to get ready when alarm rings = 55 - (1 + 2 + 3 + ... + (x-1))
Time to get ready when alarm rings = 55 - [(x^2 - x)/2]
Time to get ready when alarm rings = 55 - (x^2 - x)/2
Time to get ready when alarm rings = (2(55) - x^2 + x)/2
To find out how much time Daisy will have to get ready on the 29th day, we can substitute 29 for x in the expression:
Time to get ready = (2(55) - x^2 + x)/2
Time to get ready = (2(55) - 29^2 + 29)/2
Time to get ready = (110 - 841 + 29)/2
Time to get ready = -351/2 (since the answer is negative, Daisy will not have any time to get ready)
To find out how many days it has been when Daisy is down to sixteen minutes to get ready, we can substitute 16 for the time to get ready in the expression:
16 = (2(55) - x^2 + x)/2
32 = 110 - x^2 + x
x^2 - x - 78 = 0
(x - 9)(x + 8) = 0
Since the number of days cannot be negative, we have:
x = 9
Therefore, when Daisy is down to sixteen minutes to get ready, it has been nine days.
How much time will she have to get ready after the alarm rings on the 29th day?
When Daisy is down to sixteen minutes
I apologize for the previous response, as an AI language model I made a mistake in understanding the problem. Here's the correct solution:
To find out how much time Daisy will have to get ready on the 29th day, we need to find out how many minutes she will stay in bed on the 29th day and subtract that from the total time she has to get ready.
From the problem statement, we know that Daisy stays in bed for one more minute each day than the previous day. So, on the 29th day, she will stay in bed for 29 minutes.
The total time Daisy has to get ready is 55 minutes minus the total number of minutes she stays in bed. To find the total number of minutes she stays in bed over 29 days, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(a1 + an)
where Sn is the sum of the first n terms, a1 is the first term, and an is the nth term.
In this case, the first term is 1 (the number of minutes she stays in bed on the first day) and the common difference is 1 (she stays in bed for one more minute each day). So, the nth term is (n-1) + 1 = n.
Substituting these values into the formula, we get:
Sn = (n/2)(a1 + an)
Sn = (29/2)(1 + 29)
Sn = 435
So, over 29 days, Daisy stays in bed for a total of 435 minutes. Therefore, her total time to get ready on the 29th day is:
Time to get ready = 55 - 435
Time to get ready = -380
Since the answer is negative, Daisy will not have any time to get ready on the 29th day.
To answer the second question, when Daisy is down to sixteen minutes to get ready, we can use the same formula and solve for the number of days:
Time to get ready = 55 - (1 + 2 + 3 + ... + (n-1))
16 = 55 - (n-1)n/2
n^2 - n - 78 = 0
(n - 8)(n + 9) = 0
Since the number of days cannot be negative, the answer is:
n = 8
Therefore, when Daisy has sixteen minutes to get ready, it has been eight days.
How much time will she have to get ready after the alarm rings on the 29th day?
When Daisy is down to sixteen minutes to get ready, how many days has it been?
To find out how much time Daisy will have to get ready on the 29th day, we need to find out how many minutes she will stay in bed on the 29th day and subtract that from the total time she has to get ready.
From the problem statement, we know that Daisy stays in bed for one more minute each day than the previous day. So, on the 29th day, she will stay in bed for 29 minutes.
The total time Daisy has to get ready is 55 minutes minus the total number of minutes she stays in bed. To find the total number of minutes she stays in bed over 29 days, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(a1 + an)
where Sn is the sum of the first n terms, a1 is the first term, and an is the nth term.
In this case, the first term is 1 (the number of minutes she stays in bed on the first day) and the common difference is 1 (she stays in bed for one more minute each day). So, the nth term is (n-1) + 1 = n.
Substituting these values into the formula, we get:
Sn = (n/2)(a1 + an)
Sn = (29/2)(1 + 29)
Sn = 435
So, over 29 days, Daisy stays in bed for a total of 435 minutes. Therefore, her total time to get ready on the 29th day is:
Time to get ready = 55 - 435
Time to get ready = -380
Since the answer is negative, Daisy will not have any time to get ready on the 29th day.
To find out how many days it has been when Daisy is down to sixteen minutes to get ready, we can use the same formula and solve for the number of days:
Time to get ready = 55 - (1 + 2 + 3 + ... + (n-1))
16 = 55 - (n-1)n/2
n^2 - n - 78 = 0
(n - 8)(n + 9) = 0
Since the number of days cannot be negative, the answer is:
n = 8
Therefore, when Daisy has sixteen minutes to get ready, it has been eight days.