Question

The table below shows the profit, P(x), in dollars, from sellingx items. 2 3 6 9 14 P(x) 88.9 108 117.3 86.4 -32.7 -427.2 Using your calculator to do a quadratic regression, express the profit as a function of the number of items sold. P(x) = Round all numbers to 1 decimal place. Preview Using your quadratic regression, estimate the profit from selling 8 items? Round to 2 decimal places. Select an answer Using your quadratic regression, estimate the number of items that need to be sold to make a profit of $-328.7? Select an answer dollars items

Answer 1

A scatterplot of the given data is attached. The scatterplot resembles a downwards opening parabola so that the given darte represents a quadratic function.

| 100 , 2468 10 12 14

Let the quadratic function be y = P(x) = ax²+bx+c.

On substituting x = 1, we get a+b+c = 88.9…(1).

On substituting x = 3, we get 9a+3b+c = 117.3…(2) and

On substituting x = 14, we get196a+14b+c = -427.2…(3)

The augmented matrix for the above linear system is A (say) =

1	1	1	88.9
9	3	1	117.3
196	14	1	-427.2

To solve the above linear system, we have to reduce A to its RREf which is

1	0	0	-4.9
0	1	0	33.8
0	0	1	60

This implies that a = -4.9, b = 33.8 and c = 60.

Hence, P(x) = -4.9x²+33.8x+60.

When x = 8, we have P(x) = -4.9*64+33.8*8 +60 = -313.6 + 270.4+60 = 16.8.

Thus, the profit from selling 8 items is $ 16.8.

Further, when P(x) = -328.7, we have -4.9x²+33.8x+60 = -328.7 or, -4.9x²+33.8x+388.7 = 0.

A graph of the function y = -4.9x²+33.8x+388.7 is attached. It may be observed that the graph crosses the X-Axis at x = -6.1 and x = 13

Since x cannot be negative, hence 13 items have to be sold for a profit of $ -328.7.

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