Question

Henry is choosing between two exercise routines. In routine #1, he burns 22 calories walking. He then runs at a rate that burns 8.5 calories per minute. In routine #2, he burns 38 calories walking. He then runs at a rate 5.3 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.

Answer 1

We want routine #1 to burn at most as many calories as routine #2. So, if we call A the amount of calories burnt during routine #1, and B the amount burnt during routine #2, we should have:

A ≤ B

Now, he burns during routine #1:

A = 22 cal + (8.5 cal/min) * t

And, during routine #2, he burns:

B = 38 cal + (5.3 cal/min) * t

Then, using these expressions for A and B, we have:

22 cal + (8.5 cal/min) * t ≤ 38 cal + (5.3 cal/min) * t

Now, in order to solve this inequality for t, we can apply a sequence of algebraic operations on both sides of this inequality, until we isolate the variable t and find its value:

22 cal + (8.5 cal/min) * t ≤ 38 cal + (5.3 cal/min) * t

22 cal + (8.5 cal/min) * t - 22 cal ≤ 38 cal + (5.3 cal/min) * t - 22 cal

(8.5 cal/min) * t ≤ 16 cal + (5.3 cal/min) * t

(8.5 cal/min) * t - (5.3 cal/min) * t ≤ 16 cal + (5.3 cal/min) * t - (5.3 cal/min) * t

(3.2 cal/min) * t ≤ 16 cal

(3.2 cal/min) * t/(3.2 cal/min) ≤ 16 cal/(3.2 cal/min)

t ≤ (16 cal) * min/(3.2 cal)

t ≤ 5 min

Therefore, routine #1 burns at most as many calories as routine #2 for

t ≤ 5 min