A painter needs 3 days to paint the walls of a room. How long would it take him, working at the same rate, to paint a room that is twice as long, twice as wide, and twice as high as the original room?
The area of all the walls scales with the square of the linear dimension, as they remain similar in shape but increase in size. Doubling all the linear dimensions multiplies tha total area by 4. Painting the enlarged room therefore takes 4 x 3 = 12 days.
At an automotive repaire shop, 3.5 h of labour costs $311.50.
a) What is the labour charge a 5-h job?
b) How could you check you answer?
it would take 24 days
To find out how long it would take the painter to paint the larger room, we can calculate the ratio of the volumes of the two rooms.
Let's assume the original room has length L, width W, and height H. The larger room will have twice the length (2L), twice the width (2W), and twice the height (2H).
The volume of the original room is V1 = L * W * H.
The volume of the larger room is V2 = (2L) * (2W) * (2H) = 8 * L * W * H.
Since the painter takes 3 days to paint the original room, we can set up a proportion:
3 days / V1 = x days / V2.
Using the volumes from above, we have:
3 / (L * W * H) = x / (8 * L * W * H).
We can simplify the equation by canceling out the common factors:
3 / 1 = x / 8.
Cross-multiplying, we get:
3 * 8 = x * 1.
x = 24.
Therefore, it would take the painter 24 days to paint the larger room, which is twice as long, twice as wide, and twice as high as the original room.