Ito is training to run a 10 kilometer race in five months. His current pace is 6.25 minutes per kilometer. He wants to constantly increase his speed each month until he is running a 5.5 minute kilometer.
Ito needs to increase his pace by ____ each month.
Complete the equation to represent Ito's speed where x is the number of the training month and y is his speed in kilometers per hour. y = ___ x + ____
To find the rate of increase in Ito's pace each month, we need to calculate the difference between his current pace and his goal pace and divide it by the number of months.
The difference in pace is:
5.5 minutes/km - 6.25 minutes/km = -0.75 minutes/km
Since we want the equation to represent an increase in pace, we take the absolute value:
|-0.75 minutes/km| = 0.75 minutes/km
Now, to calculate the increase in pace each month, we divide this value by the number of months:
0.75 minutes/km ÷ 5 months = 0.15 minutes/km per month
To convert the pace to kilometers per hour, we need to find the reciprocal of the time:
1 minute/km ÷ 60 minutes/hour = 1/60 km/hour
To represent Ito's speed (y) in terms of the training month (x), we can use the formula:
y = (0.15 x + 6.25) km/hour
So, the complete equation representing Ito's speed is:
y = 0.15x + 6.25 km/hour
To find the amount by which Ito needs to increase his pace each month, we need to calculate the difference between his target pace and his current pace, and then divide it by the number of months he has to achieve his goal.
The difference between his target pace of 5.5 minutes per kilometer and his current pace of 6.25 minutes per kilometer is:
5.5 - 6.25 = -0.75
Since Ito wants to increase his speed, we consider the positive value of the difference, which is 0.75.
Now, we divide this difference by the number of months he has, which is 5, to find the amount by which he needs to increase his pace each month:
0.75 / 5 = 0.15
Therefore, Ito needs to increase his pace by 0.15 minutes per kilometer each month.
To represent Ito's speed as an equation, we can use the equation of a straight line, y = mx + c, where y is his speed in kilometers per hour, x is the number of the training month, m is the rate of increase per month, and c is the initial speed.
Since we know Ito's current pace is 6.25 minutes per kilometer, we need to convert it to kilometers per hour. There are 60 minutes in an hour, so we divide 60 by 6.25 to get the speed in kilometers per hour:
60 / 6.25 ≈ 9.6
So, Ito's initial speed is approximately 9.6 kilometers per hour.
Now, let's substitute the values into the equation:
y = 0.15x + 9.6
Therefore, the equation that represents Ito's speed is y = 0.15x + 9.6.