Question

find and simplify the ratio of the volume of figure a to the volume of figure b. the ratio of the volume of figure a to the volume of figure b is square for all xsquare (simplify your answers.)

Explanation:

Step1: Calculate the volume of Figure A

The volume formula for a rectangular prism is \(V = l\times w\times h\). For Figure A, \(l = 3x\), \(w=x\), \(h=x + 1\). So \(V_A=3x\times x\times(x + 1)=3x^{2}(x + 1)=3x^{3}+3x^{2}\).

Step2: Calculate the volume of Figure B

For Figure B, \(l=x + 2\), \(w=x\), \(h = 2x\). So \(V_B=(x + 2)\times x\times2x=2x^{2}(x + 2)=2x^{3}+4x^{2}\).

Step3: Find the ratio of \(V_A\) to \(V_B\)

The ratio \(\frac{V_A}{V_B}=\frac{3x^{3}+3x^{2}}{2x^{3}+4x^{2}}\). Factor out \(3x^{2}\) from the numerator (\(3x^{3}+3x^{2}=3x^{2}(x + 1)\)) and \(2x^{2}\) from the denominator (\(2x^{3}+4x^{2}=2x^{2}(x + 2)\)). Then \(\frac{V_A}{V_B}=\frac{3x^{2}(x + 1)}{2x^{2}(x + 2)}=\frac{3(x + 1)}{2(x + 2)}\), where \(x
eq0\) (because if \(x = 0\), the volumes \(V_A=V_B = 0\) and the ratio is undefined in the context of non - zero volumes) and \(x
eq - 2\) (to avoid a zero denominator in \(V_B\)).

Answer:

The ratio of the volume of Figure A to the volume of Figure B is \(\frac{3(x + 1)}{2(x + 2)}\) for all \(x
eq0,-2\).