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# Algebra: Linear systems of quadratic equations and inequalities and maximizing profit.

Question One
The relationship between the load on a reel, L kilonewtons per metre (L kNm 1) and the reel diameter, x metres, is modeled by a graph consisting of two parabolic arcs, AR and BC, as shown.
Arc AR is part of the parabola L =px2 + qx +r Points D(0. 1, 2.025), E(0.2, 2.9) and F(0.3, 3.425) lie on arc AR. Setup a system of three simultaneous equations relating p. q and r. Do not solve the equations.

Question Two
An orchardist has 10 hectares of land available for growing Fuji apples and Nash pears. She estimates that each hectare of apples will cost \$5000 to establish, and each hectare of pears will cost \$20 000 to establish. She is not able to spend more than \$100 000 on establishing the orchard.
It also estimated that the profit per hectare from apples will be \$4000 and the profit per hectare from pears will be \$8000. What is the maximum profit the orchardist can expect? How many hectares of Fuji apples and Nash pears should she plant to achieve this?
(Show your workly clearly. You should explain the reason behind the answer you give.)

#### Solution Preview

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1.
D is on the arc, so we have

p*x^2+ qx + r = p*0.1^2+ 0.1*q+r = 0.01p + 0.1q+r = 2.025

so 0.01p + 0.1q+r = 2.025 (1)

E is on the arc, so we have

p*x^2+ qx + r = ...

#### Solution Summary

Linear systems of quadratic equations and inequalities and maximizing profit are investigated.

\$2.19