A rock climber climbs twenty feet up the side of a cliff and stops to recheck his equipment. He then continues to ascend the side of the cliff at a constant rate of about two feet per minute.
Define units for the time Antoine climbs after checking equipment and the distance Antoine has climbed.
If it has been one minute since Antoine checked his equipment, how far has he climbed?
Antoine reaches the top of the cliff after twenty nine minutes. How high is the cliff?
Complete the rows for the speed at which Antoine climbs and the distance Antoine climbed before checking equipment. Then, enter a variable for the time Antoine climbs after checking equipment and use this variable to write an expression for the distance Antoine has climbed.
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
Time
Distance Climbed
Unit
If it has been one minute since Antoine checked his equipment, how far has he climbed?
Question 1
Antoine reaches the top of the cliff after twenty nine minutes. How high is the cliff?
Question 2
Change in feet each minute
feet per minute
Distance already climbed
feet
Expression
Question 1: 22 feet (20 feet from initial climb + 2 feet climbed in one minute after checking equipment)
Question 2: 58 feet (29 minutes x 2 feet per minute)
| | Time | Distance Climbed |
|----------------|------------|---------------------|
| Quantity Name | units | units |
| Quantity Desc. | "after checking equipment" | "before checking equipment" |
| Units | minutes | feet |
| Algebraic Expr | (time - 1) | 2(time - 1) + 20 |
| Numeric Value | 28 | 58 |
seven yards from the half line towards the other team's goal line. The team then receives some penalties. Each penalty moves the team ten yards away from the other team's goal line. Measuring from the half line, positions closer to the opponent's goal line are positive values. Positions closer to the team's own goal line are negative values. Penalties are negative values.
Define a unit for the position relative to the half line.
If the team gets six penalties, what is their position relative to the half line?
If the team gets seven penalties, what is their position relative to the half line?
Complete the rows for the change in position for each penalty and the team's initial position compared to the half line. Then, enter a variable for the number of penalties and use this variable to write an expression for the position relative to the half line.
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
Penalties
Position Relative to Half Line
Unit
penalties
If the team gets six penalties, what is their position relative to the half line?
Question 1
If the team gets seven penalties, what is their position relative to the half line?
Question 2
Change in yards for each penalty
yards per penalty
Position of the team before the penaties
yards
Expression
Question 1: -40 yards (-7 penalties x 10 yards per penalty + 7 yards initial position)
Question 2: -50 yards (-7 penalties x 10 yards per penalty + 7 yards initial position)
| | Penalties | Position Relative to Half Line |
|-----------------|-----------|--------------------------------|
| Quantity Name | units | units |
| Quantity Desc. | "number of penalties" | "position before penalties" |
| Units | penalties | yards |
| Algebraic Expr | -10(# of penalties) + initial position | -10(# of penalties) + initial position |
| Numeric Value | 6 | -23 |
To find the distance Antoine has climbed after checking his equipment, we need to add the distance he climbed before checking his equipment (which is given as 20 feet) to the product of his climbing rate (2 feet per minute) and the time he has climbed after checking his equipment.
Let's define the units:
- Time: minutes
- Distance climbed: feet
So, if it has been one minute since Antoine checked his equipment, we can substitute this value into the expression:
Distance climbed = 20 feet + (2 feet/minute) * 1 minute
Distance climbed = 20 feet + 2 feet
Distance climbed = 22 feet
Therefore, if it has been one minute since Antoine checked his equipment, he has climbed a distance of 22 feet.
To find the height of the cliff, we know that Antoine reaches the top after twenty-nine minutes. We can simply multiply his climbing rate (2 feet per minute) by the total time he has climbed:
Height of the cliff = (2 feet/minute) * 29 minutes
Height of the cliff = 58 feet
Therefore, the height of the cliff is 58 feet.