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malfutka [58]
3 years ago
5

The top and bottom margins of a poster are each 15 cm and the side margins are each 10 cm. If the area of printed material on th

e poster is fixed at 2400 cm2, find the dimensions of the poster with the smallest area.
Mathematics
1 answer:
12345 [234]3 years ago
4 0

Answer:

the dimension of the poster = 90 cm length and 60 cm  width i.e 90 cm by 60 cm.

Step-by-step explanation:

From the given question.

Let p be the length of the of the printed material

Let q be the width of the of the printed material

Therefore pq = 2400 cm ²

q = \dfrac{2400 \ cm^2}{p}

To find the dimensions of the poster; we have:

the length of the poster to be p+30 and the width to be \dfrac{2400 \ cm^2}{p} + 20

The area of the printed material can now be:  A = (p+30)(\dfrac{2400 }{p} + 20)

=2400 +20 p +\dfrac{72000}{p}+600

Let differentiate with respect to p; we have

\dfrac{dA}{dp}= 20 - \dfrac{72000}{p^3}

Also;

\dfrac{d^2A}{dp^2}= \dfrac{144000}{p^3}

For the smallest area \dfrac{dA}{dp }=0

20 - \dfrac{72000}{p^2}=0

p^2 = \dfrac{72000}{20}

p² = 3600

p =√3600

p = 60

Since p = 60 ; replace p = 60 in the expression  q = \dfrac{2400 \ cm^2}{p}   to solve for q;

q = \dfrac{2400 \ cm^2}{p}

q = \dfrac{2400 \ cm^2}{60}

q = 40

Thus; the printed material has the length of 60 cm and the width of 40cm

the length of the poster = p+30 = 60 +30 = 90 cm

the width of the poster = \dfrac{2400 \ cm^2}{p} + 20 = \dfrac{2400 \ cm^2}{60} + 20  = 40 + 20 = 60

Hence; the dimension of the poster = 90 cm length and 60 cm  width i.e 90 cm by 60 cm.

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