Math

QuestionПравоъгьлен паралелепипед с размери a,b,ca, b, c има отношение a:b=4:3a:b=4:3 и b:c=2:5b:c=2:5. Сборът на ръбовете е 232 cm232 \mathrm{~cm}. Какъв е обемът му?

Studdy Solution

STEP 1

Assumptions1. The dimensions of the rectangular parallelepiped are a,ba, b and cc . The ratio of aa to bb is 4343
3. The ratio of bb to cc is 55
4. The sum of all edges of the parallelepiped is 232cm232 \, cm
5. We need to find the volume of the parallelepiped in cubic centimeters

STEP 2

First, we need to express the dimensions a,ba, b and cc in terms of the same variable. From the given ratios, we can express aa and cc in terms of bb.
a=4ba = \frac{4}{}bc=52bc = \frac{5}{2}b

STEP 3

The sum of all edges of a rectangular parallelepiped is (a+b+c)(a+b+c). We can substitute the expressions for aa and cc from step2 into this equation.
(a+b+c)=(3b+b+52b)(a+b+c) =\left(\frac{}{3}b+b+\frac{5}{2}b\right)

STEP 4

implify the equation to find an expression for bb.
4(43b+b+2b)=4(236b)4\left(\frac{4}{3}b+b+\frac{}{2}b\right) =4\left(\frac{23}{6}b\right)236b=2324\frac{23}{6}b = \frac{232}{4}

STEP 5

olve the equation to find the value of bb.
b=232×4×23b = \frac{232 \times}{4 \times23}

STEP 6

Calculate the value of bb.
b=12cmb =12 \, cm

STEP 7

Substitute the value of bb into the expressions for aa and cc to find their values.
a=43×12cma = \frac{4}{3} \times12 \, cmc=52×12cmc = \frac{5}{2} \times12 \, cm

STEP 8

Calculate the values of aa and cc.
a=16cma =16 \, cmc=30cmc =30 \, cm

STEP 9

Now that we have the dimensions of the parallelepiped, we can calculate its volume. The volume VV of a rectangular parallelepiped is given by the formula V=abcV = abc.
V=a×b×cV = a \times b \times c

STEP 10

Substitute the values of aa, bb and cc into the formula to calculate the volume.
V=16cm×12cm×30cmV =16 \, cm \times12 \, cm \times30 \, cm

STEP 11

Calculate the volume of the parallelepiped.
V=5760cm3V =5760 \, cm^3The volume of the parallelepiped is5760 cubic centimeters.

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