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natima [27]
2 years ago
12

Cynthia Besch wants to buy a rug for a room that is 24 ft wide and 33 ft long. She

Mathematics
1 answer:
qaws [65]2 years ago
5 0
<h3>Answer:   16 ft by 25 ft</h3>

=========================================================

Explanation:

Refer to the diagram below.

The outer dimensions 24 ft and 33 ft shrink down to 24-2x ft and 33-2x ft respectively. This subtraction of 2x is due to subtracting two copies of x per side.

The carpet has area of (24-2x)(33-2x)

Cynthia can afford to buy 400 sq ft of carpet

So we set 400 equal to that previous expression and solve for x

(24-2x)(33-2x) = 400

24(33-2x) - 2x(33-2x) = 400

792-48x - 66x + 4x^2 = 400

4x^2 - 114x + 792 = 400

4x^2 - 114x + 792-400 = 0

4x^2 - 114x + 392 = 0

From here, use the quadratic formula to isolate x.

Plug in a = 4, b = -114, c = 392

x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-(-114)\pm\sqrt{(-114)^2-4(4)(392)}}{2(4)}\\\\x = \frac{114\pm\sqrt{6724}}{8}\\\\x = \frac{114\pm82}{8}\\\\x = \frac{114+82}{8} \ \text{ or } \ x = \frac{114-82}{8}\\\\x = \frac{196}{8} \ \text{ or } \ x = \frac{32}{8}\\\\x = 24.5 \ \text{ or } \ x = 4\\\\

The two possible solutions are x = 24.5 and x = 4

But if x = 24.5, then 24 - 2x = 24 - 2*24.5 = -25 which isn't possible. We cannot have a negative width or negative length for the carpet.

Luckily x = 4 does work since

24 - 2x = 24 - 2*4 = 16

33 - 2x = 33 - 2*4 = 25

Both results are positive.

Therefore, the carpet has dimensions of <u>16 ft by 25 ft</u>

Check: 16*25 = 400, so the answer is confirmed

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The isosceles trapezoids, ABCD and EFGH, are similar quadrilaterals. The scale factor between the trapezoids is 2:3, = 6 centime
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Answer:

Part A) see the explanation

Part B) see the explanation

Part C) The perimeter of trapezoid ABCD is 32 centimeters and the perimeter of trapezoid EFGH is 48 centimeters

Step-by-step explanation:

<u><em>The complete question is</em></u>

The isosceles trapezoids, ABCD and EFGH, are similar quadrilaterals. The scale factor between the trapezoids is 2:3, GH = 6 centimeters, AD = 8 centimeters, and AB is three times the length of DC

Part A) Write a similarity statement for each of the four pair of corresponding sides

we know that

If two figures are similar, then the ratio of its corresponding sides is proportional

The corresponding sides are

AB and EF

BC and FG

CD and GH

AD and EH

so

\frac{AB}{EF}=\frac{BC}{FG}=\frac{CD}{GH}=\frac{AD}{EH}

Part B) Write a congruence statement for each of the four pair of corresponding angles.

we know that

If two figures are similar, then its corresponding angles are congruent

The corresponding angles are

∠A and ∠E

∠B and ∠F

∠C and ∠G

∠D and ∠H

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∠A ≅ ∠E

∠B ≅ ∠F

∠C ≅ ∠G

∠D ≅ ∠H

Part C) Determine the perimeter for each of the isosceles trapezoids

we have

The scale factor between the trapezoids is 2:3, GH = 6 centimeters, AD = 8 centimeters, and AB is three times the length of DC

step 1

Find the measure of CD

Remember that

The ratio between corresponding sides is proportional and this ratio is the scale factor

Let

z ----> the scale factor

z=\frac{2}{3}

so

\frac{2}{3}=\frac{CD}{GH}}

we have

GH=6\ cm

substitute

\frac{2}{3}=\frac{CD}{6}}\\\\CD=4\ cm

step 2

Find the measure of AB

Remember that

AB is three times the length of DC

so

AB=3DC

DC=CD=4\ cm

substitute

AB=3(4)=12\ cm

step 3

Find the measure of BC

Remember that in an isosceles trapezoid, the legs are equal

so

BC=AD

we have

AD=8\ cm

therefore

BC=8\ cm

step 4

Find the perimeter of trapezoid ABCD

P=AB+BC+CD+AD

substitute the values

P=12+8+4+8=32\ cm

step 5

Find the perimeter of trapezoid EFGH

we know that

If two figures are similar, then the ratio of its perimeters is equal to the scale factor

Let

z ---> the scale factor

x ----> the perimeter of trapezoid ABCD

y ----> the perimeter of trapezoid EFGH

so

z=\frac{x}{y}

we have

z=\frac{2}{3}

x=32\ cm

substitute

\frac{2}{3}=\frac{32}{y}

y=32(3)/2=48\ cm

therefore

The perimeter of trapezoid EFGH is 48 centimeters

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