Question

An architect is redesigning a rectangular room on the blueprints of the house. He decides to double the width of the room, increase the length by \(50 \%,\) and increase the height by \(20 \% .\) By what factor has the volume of the room increased?

Short answer

Answer: The volume of the room has increased by a factor of 3.6.

Step-by-step solution

Identifying the Changes in Dimensions

The problem gives us the following information: - Width of the room doubled - Length of the room increased by \(50 \%\) - Height of the room increased by \(20 \%\) Let the initial dimensions of the room be \(w\) (width), \(l\) (length), and \(h\) (height).

Calculating the New Dimensions

Using the information given, we can calculate the new dimensions as follows: - New Width: \(2w\) - New Length: \(l + 0.5l = 1.5l\) - New Height: \(h + 0.2h = 1.2h\)

Finding the Initial and Final Volumes

Volume of a rectangular room can be calculated using the formula: \(V = w \times l \times h\) Initial volume (\(V_{initial}\)): \(V_{initial} = wlh\) Final volume (\(V_{final}\)): \(V_{final} = (2w)(1.5l)(1.2h) = 3.6wlh\)

Calculating the Increase in Volume

To find the factor by which the volume of the room has increased, we need to divide the final volume by the initial volume: Factor of increase in volume: \(\frac{V_{final}}{V_{initial}} = \frac{3.6wlh}{wlh}\)

Finding the Solution

Simplifying the expression from Step 4: Factor of increase in volume: \(\frac{3.6wlh}{wlh} = 3.6\) The volume of the room has increased by a factor of \(\boldsymbol{3.6}\).