Question

use the image to answer the question
figure a
figure b
image long description
the volume of a triangular prism with the area of the triangular base b and height h can be found using the formula v = bh. calculate and compare the volume of the two figures in the image. choose which statement is true
(1 point)
there is no scale factor stated for the volume
the volumes of the two figures have a ratio of 25:1
the volumes of the two figures have a ratio of 125:1
the volumes of the two figures have a ratio of 5:1

Explanation:

Step1: Calculate the area of the triangular base for Figure A

The formula for the area of a triangle is \(B=\frac{1}{2}bh\). For Figure A, \(b = 27.5\) ft and \(h=27.5\) ft. So, \(B_A=\frac{1}{2}\times27.5\times27.5=\frac{1}{2}\times756.25 = 378.125\) square - feet. Then, using the volume formula \(V = Bh\) (where \(h = 55\) ft for Figure A), \(V_A=378.125\times55=20796.875\) cubic - feet.

Step2: Calculate the area of the triangular base for Figure B

For Figure B, \(b = 5.5\) ft and \(h = 5.5\) ft. So, \(B_B=\frac{1}{2}\times5.5\times5.5=\frac{1}{2}\times30.25 = 15.125\) square - feet. Then, using the volume formula \(V = Bh\) (where \(h = 11\) ft for Figure B), \(V_B=15.125\times11 = 166.375\) cubic - feet.

Step3: Find the ratio of the volumes

\(\frac{V_A}{V_B}=\frac{20796.875}{166.375}\). Since \(20796.875\div166.375=\frac{20796.875\times1000}{166.375\times1000}=\frac{20796875}{166375}\). Dividing numerator and denominator by \(166375\), we get \(\frac{20796875\div166375}{166375\div166375}=125\). So the ratio \(V_A:V_B = 125:1\).

Another way:
The ratio of corresponding side lengths \(k=\frac{27.5}{5.5}=\frac{55}{11}=5\). For similar solids (these are similar triangular prisms as their corresponding sides are in proportion), the ratio of volumes \(V_1:V_2=k^3\). Since \(k = 5\), \(k^3=5^3=125\). So the ratio of volumes \(V_A:V_B=125:1\).

Answer:

The volumes of the two figures have a ratio of \(125:1\)