Dan is choosing between two exercise routines.
In routine 1, he burns 20 calories walking. He then runs at a rate that burns 10.5 calories per minute.
In routine 2, he burns 39 calories walking. He then runs at a rate that burns 5.75 calories per minute.
For what amounts of time spent running will routine 1 burn at most as many calories as routine 2? Use t for the number of minutes spent running, and solve your inequality for t.
To determine the amount of time spent running (t) for which routine 1 burns at most as many calories as routine 2, we need to set up an inequality based on the calorie burn of each routine.
In routine 1, the total calorie burn is given by:
Calories burned in walking + Calories burned in running
The calories burned in walking for routine 1 is 20, and the calories burned in running is 10.5t (where t is the time spent running in minutes). Therefore, the total calorie burn for routine 1 is: 20 + 10.5t.
Similarly, in routine 2, the total calorie burn is given by:
Calories burned in walking + Calories burned in running
The calories burned in walking for routine 2 is 39, and the calories burned in running is 5.75t. Therefore, the total calorie burn for routine 2 is: 39 + 5.75t.
To find the maximum time (t) for which routine 1 burns at most as many calories as routine 2, we can set up the following inequality:
20 + 10.5t ≤ 39 + 5.75t
Now, let's solve this inequality for t:
Subtracting 5.75t from both sides of the inequality:
20 + 10.5t - 5.75t ≤ 39
Simplifying the equation:
20 + 4.75t ≤ 39
Subtracting 20 from both sides of the equation:
4.75t ≤ 19
Dividing both sides of the equation by 4.75:
t ≤ 4
Therefore, the solution to the inequality t ≤ 4 represents the amount of time spent running in which routine 1 will burn at most as many calories as routine 2.
To solve for "as many," use this equation to solve for t.
20 + 10.5t = 39 + 5.75t