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# A rectangular circuit board is designed to have perimeter p, diagonal

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Joined: 02 Sep 2009
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A rectangular circuit board is designed to have perimeter p, diagonal  [#permalink]

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11 Sep 2018, 02:50
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Difficulty:

45% (medium)

Question Stats:

71% (01:43) correct 29% (02:28) wrong based on 21 sessions

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A rectangular circuit board is designed to have perimeter p, diagonal d and area k. Which of the following equations must be true?

A. $$d^2 - p2 + 2k = 0$$

B. $$2d^2 - p2 + 2k = 0$$

C. $$4d^2 - p2 + 4k = 0$$

D. $$4d^2 - p2 + 8k = 0$$

E. $$4d^2 -2p2 + 8k = 0$$

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A rectangular circuit board is designed to have perimeter p, diagonal  [#permalink]

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11 Sep 2018, 05:17
Bunuel wrote:
A rectangular circuit board is designed to have perimeter p, diagonal d and area k. Which of the following equations must be true?

A. $$d^2 - p2 + 2k = 0$$

B. $$2d^2 - p2 + 2k = 0$$

C. $$4d^2 - p2 + 4k = 0$$

D. $$4d^2 - p2 + 8k = 0$$

E. $$4d^2 -2p2 + 8k = 0$$

Let, l and b be the dimensions of rectangle

Area = $$l*b = k$$
Perimeter $$= 2(l+b) = p$$
i.e. $$(l+b) = p/2$$
Diagonal = $$√(l^2+b^2) = d$$
i.e. $$(l^2+b^2) = d^2$$

Now, $$(l+b)^2 = l^2+b^2 +2*l*b$$
i.e. $$(p/2)^2 = d^2 +2*k$$

i.e. $$(p)^2 = 4d^2 +8*k$$

i.e. $$4d^2 - p^2 + 8k = 0$$

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Re: A rectangular circuit board is designed to have perimeter p, diagonal  [#permalink]

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14 Sep 2018, 17:52
Bunuel wrote:
A rectangular circuit board is designed to have perimeter p, diagonal d and area k. Which of the following equations must be true?

A. $$d^2 - p2 + 2k = 0$$

B. $$2d^2 - p2 + 2k = 0$$

C. $$4d^2 - p2 + 4k = 0$$

D. $$4d^2 - p2 + 8k = 0$$

E. $$4d^2 -2p2 + 8k = 0$$

(Note: p2 should be p^2.)

Let the length and width of the rectangular circuit board be L and W, respectively. So we have

2(L + W) = p, L^2 + W^2 = d^2 and LW = k

Since L + W = p/2, we have

(L + W)^2 = (p/2)^2

L^2 + W^2 + 2LW = p^2/4

Since L^2 + W^2 = d^2 and LW = k, we have:

d^2 + 2k = p^2/4

4d^2 + 8k = p^2

4d^2 + 8k - p^2 = 0

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Re: A rectangular circuit board is designed to have perimeter p, diagonal   [#permalink] 14 Sep 2018, 17:52
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