How to find the minimum value of y?The diagram http://i49.tinypic.com/20sw044.jpg shows a rectangle ABCD. The lengths of the sides AB and DA are 12cm and 9cm respectively. P and Q are points on...

How to find the minimum value of y?

The diagram http://i49.tinypic.com/20sw044.jpg shows a rectangle ABCD. The lengths of the sides AB and DA are 12cm and 9cm respectively. P and Q are points on the sides DC and CB respectively such that the angle APQ is a right angle. If DP=x cm and QB=y cm, how to express y in terms of x? (Answer: 1/9(x^2 - 12x + 81) And then how to find the minimum value of y (Answer: 5cm) ? 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Notice that you may use Pythagorean theorem in the right angle triangle PCQ such that:

`PQ^2 = PC^2 + CQ^2`

Notice that you may evaluate PC such that:

`PC = 12 - x`

You may also evaluate CQ such that:

`CQ = 9 - y`

`PQ^2 = (12 - x)^2 + (9 - y)^2`

Opening the brackets yields:

`PQ^2 = 144 - 24x + x^2 + 81 - 18y + y^2`

You also may consider PQ as the leg of right angle triangle APQ such that:

`PQ^2 = AQ^2 - AP^2`

You may find `AQ^2`  using Pythagorean theorem in the right angle triangle ABQ such that:

`AQ^2 = 144 + y^2`

You may find `AP^2`  using Pythagorean theorem in the right angle triangle ADP such that:

`AP^2 = 81 + x^2`

`PQ^2 = 144 + y^2 - 81 - x^2`

`PQ^2 = 63 + y^2 - x^2`

Setting the equations `PQ^2 = 63 + y^2 - x^2`  and `PQ^2 = 144 - 24x + x^2 + 81 - 18y + y^2`  equal yields:

`63 + y^2 - x^2 = 144 - 24x + x^2 + 81 - 18y + y^2`

`2x^2 + 162 - 24x - 18y = 0`

You need to expres y in terms of x, hence, you need to isolate y to the left side such that:

`-18y = 24x - 2x^2 - 162`

Dividing by -2 yields:

`9y = x^2 - 12x + 81 => y = (x^2 - 12x + 81)/9`

You need to find the minimum value of y, hence, you need to differentiate y with respect to x such that:

`dy/dx = (2x-12)/9`

Putting `dy/dx = 0`  yields:

`2x-12 = 9 => 2x = 31 => x = 31/2 => x = 15.5`

Substituting 15.5 for x in equation of y yields the minimum value of y such that:

`y = (240.25 - 186 + 81)/9 ~~ 15`

Hence, expressing y in terms of x yields `y = (x^2 - 12x + 81)/9`  and evaluating the minimum value of y yields `y~~ 15.`

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