Question

the scale factor of a large oil container to a small oil container is 0.075. the large oil container can carry 144,000 cm³ of oil.
how much oil can the small container carry?
0.075 cm³
60.75 cm³
810 cm³
10,800 cm³

Explanation:

Step1: Recall volume scale factor rule

For similar solids, the ratio of volumes is the cube of the scale factor. But here, the scale factor from large to small is 0.075, so volume of small = volume of large × (scale factor)³? Wait, no—wait, scale factor of large to small is 0.075, meaning small is 0.075 times the linear dimensions of large? Wait, no, scale factor \( k \) for linear dimensions: if large to small is \( k = 0.075 \), then volume scale factor is \( k^3 \)? Wait, no, wait: actually, if the scale factor (linear) of large to small is \( k \), then the volume of small is volume of large × \( k^3 \)? Wait, no, let's think again. Let’s denote: Let \( V_{large} \) be volume of large, \( V_{small} \) volume of small. The linear scale factor from large to small is \( k = 0.075 \). Then, since volume is a three - dimensional measurement, \( V_{small}=V_{large}\times k^{3}\)? Wait, no, that would be if we were going from small to large. Wait, no: if the large is scaled down by a factor of \( k \) (linear) to get the small, then the volume of the small is \( V_{large}\times k^{3} \). Wait, let's check the logic. Suppose the linear dimensions of large are \( L \), \( W \), \( H \), so volume \( V_{large}=L\times W\times H \). The small has linear dimensions \( kL \), \( kW \), \( kH \), so volume \( V_{small}=kL\times kW\times kH=k^{3}LWH = k^{3}V_{large} \). So yes, \( V_{small}=V_{large}\times k^{3} \), where \( k \) is the linear scale factor from large to small.

Step2: Calculate the volume of the small container

Given \( V_{large} = 144000\space cm^{3} \), \( k = 0.075 \). Then \( V_{small}=144000\times(0.075)^{3} \). First, calculate \( 0.075^{3}=0.075\times0.075\times0.075 \). \( 0.075\times0.075 = 0.005625 \), then \( 0.005625\times0.075=0.000421875 \). Then \( V_{small}=144000\times0.000421875 \). \( 144000\times0.000421875 = 144000\times\frac{421875}{1000000000}=\frac{144000\times421875}{1000000000} \). But it's easier to calculate \( 144000\times0.000421875 \). \( 144000\times0.000421875=144000\times\frac{421875}{1000000000} \). Alternatively, \( 144000\times0.075 = 10800 \), \( 10800\times0.075 = 810 \), \( 810\times0.075 = 60.75 \)? Wait, no, that's wrong. Wait, no, I messed up the scale factor. Wait, maybe the scale factor is from small to large? Wait, the problem says "the scale factor of a large oil container to a small oil container is 0.075". So large to small: linear scale factor \( k = 0.075 \), so volume scale factor is \( k^{3} \). Wait, but let's re - examine. Wait, maybe I got the direction wrong. If the scale factor of large to small is 0.075, that means small is 0.075 times the linear size of large. So volume of small is (0.075)³ times volume of large. But let's compute (0.075)³:

\( 0.075\times0.075 = 0.005625 \)

\( 0.005625\times0.075=0.000421875 \)

Then \( 144000\times0.000421875 = 60.75 \)? Wait, no, that's not matching the options. Wait, maybe the scale factor is the ratio of small to large? Wait, maybe the problem means that the scale factor (linear) of small to large is 0.075? No, the problem says "scale factor of a large oil container to a small oil container is 0.075". So large : small (linear) = 1:0.075? No, scale factor is defined as (image size)/(original size). So if large is the original and small is the image, then scale factor \( k=\frac{\text{small linear dimension}}{\text{large linear dimension}} = 0.075 \). Then volume scale factor is \( k^{3} \), so volume of small is \( V_{large}\times k^{3} \). But let's check the options. Wait, maybe I made a mistake in the volume scale factor. Wait, maybe the scale factor here is for volume? No, scale factor for similar solids: linear scale factor \( k \), area scale factor \( k^{2} \), volume scale factor \( k^{3} \).

Wait, let's recalculate:

\( 0.075^{3}=0.075\times0.075\times0.075 \)

\( 0.075\times0.075 = 0.005625 \)

\( 0.005625\times0.075 = 0.000421875 \)

\( 144000\times0.000421875=144000\times\frac{421875}{1000000000}=\frac{144000\times421875}{1000000000} \)

\( 144000\times421875 = 144000\times421875 = 60750000000 \)

\( 60750000000\div1000000000 = 60.75 \). Wait, but 60.75 is one of the options. Wait, but let's think again. Maybe the scale factor is the ratio of the small to the large in volume? No, the problem says "scale factor", which for similar solids is linear unless stated otherwise. But maybe in this problem, they consider the scale factor for volume? Wait, no, that would be incorrect. But let's check the numbers. If we assume that the scale factor is for volume (which is wrong, but let's see), then volume of small = 144000×0.075 = 10800, which is an option, but that's wrong. If we assume that the scale factor is linear, and we made a mistake in the direction. Wait, maybe the scale factor is large to small in linear, but we need to do volume of small = volume of large × (scale factor)³? Wait, no, if large is scaled down by a factor of \( k \) (linear) to get small, then small volume is large volume × \( k^{3} \). But let's check the calculation again:

\( 0.075^3 = 0.075\times0.075\times0.075 = 0.000421875 \)

\( 144000\times0.000421875 = 60.75 \)

Yes, that's correct. So the small container can carry \( 60.75\space cm^{3} \) of oil.

Answer:

\( 60.75\space cm^{3} \) (corresponding to the option "60.75 \( cm^{3} \)")