Question Video: Finding the Average Rate of Change of a Rational Function in a Real-World Context | Nagwa Question Video: Finding the Average Rate of Change of a Rational Function in a Real-World Context | Nagwa

Question Video: Finding the Average Rate of Change of a Rational Function in a Real-World Context Mathematics

A farm’s production in kilograms 𝑦 as a function of the kilograms of insecticide 𝑥 is given by 𝑦 = 146 − (47/(3𝑥 + 8)). Find the average rate of change in 𝑦 when 𝑥 varies from 13 to 17.

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Video Transcript

A farm’s production in kilograms 𝑦 as a function of the kilograms of insecticide 𝑥 is given by 𝑦 is equal to 146 minus 47 divided by three 𝑥 plus eight. Find the average rate of change in 𝑦 when 𝑥 varies from 13 to 17.

The question gives us a function of 𝑦 in terms of 𝑥. And it wants us to find the average rate of change of 𝑦 when 𝑥 varies from 13 to 17. And we recall the average rate of change of a function 𝑓 of 𝑥 from 𝑥 is equal to 𝑎 to 𝑥 is equal to 𝑏 is given by 𝑓 average is equal to 𝑓 evaluated at 𝑏 minus 𝑓 evaluated at 𝑎 divided by 𝑏 minus 𝑎. Since the question wants us to find the average rate of change of 𝑦, where 𝑦 is equal to 146 minus 47 divided by three 𝑥 plus eight, when 𝑥 varies from 13 to 17. We’ll set our function 𝑓 of 𝑥 equal to 146 minus 47 divided by three 𝑥 plus eight, 𝑎 equal to 13 and 𝑏 equal to 17.

To find the average rate of change in this case, we’ll start by evaluating 𝑓 at 13 and 𝑓 at 17. Substituting 𝑥 is equal to 13 gives us 𝑓 of 13 is 146 minus 47 divided by three times 13 plus eight. We can simplify this. Three times 13 is 39 and then we add eight. This gives us 47. And 47 divided by 47 simplifies to give us one. And, of course, 146 minus one is equal to 145. We can then do the same for 𝑓 evaluated at 17. We’ll get 146 minus 47 divided by three times 17 plus eight. We can simplify this. Three times 17 is 51 and then we add eight. And this gives us 59.

We’re now ready to calculate the average rate of change of our function. We have 𝑓 average is equal to 𝑓 evaluated at 17 minus 𝑓 evaluated at 13 divided by 17 minus 13. We’ve already calculated 𝑓 evaluated at 17 and 𝑓 evaluated at 13. This gives us the numerator of 146 minus 47 divided by 59, and then we subtract 145. And in our denominator, we have 17 minus 13 is four. We can simplify our numerator. We have 146 minus 145 is equal to one. So we have that 𝑓 average is equal to one minus 47 over 59 all divided by four.

We can simplify this further by rewriting one as 59 divided by 59. We can then simply subtract the fractions in our numerator. This gives us 12 over 59 all divided by four. Instead of dividing by four, we can multiply by the reciprocal of four. This gives us 12 over 59 multiplied by a quarter. And then, we see that 12 divided by four is just equal to three. Giving us that 𝑓 average was equal to three divided by 59. Therefore, we’ve shown if 𝑦 is equal to 146 minus 47 divided by three 𝑥 plus eight, then the average rate of change of 𝑦 when 𝑥 varies from 13 to 17 is three divided by 59.

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