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Alborosie
1 month ago
12

A rectangle has area 25 m2. express the perimeter of the rectangle as a function of the length l of one of its sides. p(l) = sta

te the domain of p. (assume the length of the rectangle is larger than its width. enter your answer using interval notation.)
Mathematics
1 answer:
zzz [12.3K]1 month ago
4 0
a) The area of a rectangle is calculated by multiplying the length by the width:
   A = l·w

This formula allows us to express the width in terms of area and length. By dividing A by l, we find
   A/l = w

We also recognize that the perimeter of a rectangle is the total length around it.
   P = l + w + l + w
   P = 2(l + w)
We aim to rewrite the perimeter formula to isolate l on one side, using the expression for w derived earlier.
   P = 2(l + A/l)

By substituting the known value for A, we can express p(l) as
   p(l) = 2(l + 25/l)
   p(l) = 2l + 50/l


b)
For lengths exceeding widths, we have
   l > w
   l > A/l
   l² > A
   l > √A
   l > √25

This indicates that the domain of p(l) is
   l ∈ (5, ∞)..... meters
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B) Maia had 2544 beads at first.

Step-by-step analysis:

Let x denote the beads with Jenny and y for Maia.

The information provided states that Maia possesses 2064 beads more than Jenny, which can be represented mathematically as:

y=x+2064...(1)

Additionally, after Maia made a necklace with 144 beads, she had five times more beads than Jenny.

This can also be formulated as an equation:

y-144=5x...(2)

A) Since Maia had 2064 beads more than Jenny before using 144 beads, we calculate her final bead count by subtracting 144 from 2064.

\text{Number of beads Maia had more than Jenny in the end}=2064-144

\text{Number of beads Maia had more than Jenny in the end}=1920

Thus, Maia is left with 1920 beads more than Jenny.

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By inserting equation (1) into equation (2), we arrive at:

x+2064-144=5x

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x-x+1920=5x-x

1920=4x

Now, let's divide our equation by 4.

\frac{1920}{4}=\frac{4x}{4}

480=x

Now, we'll substitute x=480 into equation (1) to isolate y.

y=480+2064

y=2544

Conclusively, Maia initially had 2544 beads.

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