Video: Finding the Point That Maximizes the Objective Function given the Graph of the Constrains

Determine the values of π‘₯ and 𝑦 that maximize the function 𝑝 = 5π‘₯ + 2𝑦. Write your answer as a point (π‘₯, 𝑦).

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Video Transcript

Determine the values of π‘₯ and 𝑦 that maximise the function 𝑝 equals five π‘₯ plus two 𝑦. Write your answer as a point π‘₯, 𝑦.

We’ve been given a region which represents a series of inequalities. The topmost horizontal inequality is 𝑦 is less than or equal to eight. The vertical inequality is π‘₯ is less than or equal to seven. The region is also bound by the π‘₯-axis. So 𝑦 must be greater than or equal to zero. And finally, we have a diagonal line with a 𝑦-intercept at eight and a gradient of negative eight over three.

This inequality is, therefore, 𝑦 is greater than or equal to negative eight over three π‘₯ plus eight. All maxima and minima occur at the corners of the region. We’re specifically interested in the maximum of the function. So by finding the value of π‘₯ and 𝑦 at these points, we can determine which order pair when substituted into 𝑝 equals five π‘₯ plus two 𝑦 will give us the largest possible value for 𝑝.

Our vertices fall at seven, zero; seven, eight; zero, eight; and three, zero. At zero, eight, 𝑝 is equal to five times zero, add two times eight, which gives us a value of 16. At three, zero, 𝑝 is equal to five times three, add two times zero, which is 15. At seven, eight, 𝑝 is equal to five times seven, add two times eight, which is 51. And at seven, zero, 𝑝 is equal to five times seven, add two times zero, which is 35.

Our function 𝑝 equals five π‘₯ plus two 𝑦 is therefore greatest when π‘₯ equals seven and 𝑦 equals eight. The values of π‘₯ and 𝑦 that maximise our function are seven, eight.

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