Suppose you build an architectural model of a new concert hall using a scale factor of 34. How will the volume of the actual concert hall compare to the volume of the square model?
The volume of two similar objects is proportional to the cube of their respective sides, so
volume of the real thing : Volume of model = 34^3 : 1^3 = 39,304 : 1
Well, first of all, I must say, if you're building a concert hall using a scale factor of 34, you better make sure you have a massive backyard! Now, to answer your question, when we talk about scaling, volume is actually cubed. So if you have a scale factor of 34, the volume of the actual concert hall would be 34 cubed times larger than the volume of the square model. That's a whole lot of extra space for all those concertgoers to rock out!
To determine how the volume of the actual concert hall compares to the volume of the square model, we need to consider the relationship between the scale factor and the three-dimensional measurements of the model and the actual concert hall.
Since the scale factor is 34, it means that every dimension of the model is 34 times smaller than the corresponding dimension of the actual concert hall.
Let's assume that the length, width, and height of the square model are represented by L_model, W_model, and H_model respectively. Similarly, the length, width, and height of the actual concert hall are represented by L_actual, W_actual, and H_actual respectively.
Since the square model is a three-dimensional shape, we can calculate its volume using the formula: Volume_model = L_model * W_model * H_model.
The volume of the actual concert hall can be calculated similarly: Volume_actual = L_actual * W_actual * H_actual.
Given that the scale factor is 34, we can express the dimensions of the model in terms of the actual concert hall dimensions: L_actual = L_model * 34, W_actual = W_model * 34, H_actual = H_model * 34.
Substituting these values into the volume formulas, we get:
Volume_model = L_model * W_model * H_model
= (L_actual / 34) * (W_actual / 34) * (H_actual / 34)
= (L_actual * W_actual * H_actual) / (34 * 34 * 34)
= (L_actual * W_actual * H_actual) / (34^3)
Therefore, the volume of the square model is equal to the volume of the actual concert hall divided by (34^3).
In summary, the volume of the actual concert hall will be 34^3 times greater than the volume of the square model.
To determine how the volume of the actual concert hall compares to the volume of the square model, we need to understand how the scale factor relates to volume. Volume is a three-dimensional measurement, so it is affected by scaling in all three dimensions: length, width, and height.
Let's assume that the length, width, and height of the square model are all being scaled up by a factor of 34.
The scale factor applies to all three dimensions of the model, which means the length, width, and height will be multiplied by 34.
The volume of the square model can be calculated by multiplying the length, width, and height together.
Now, let's say the volume of the square model is V_square.
So, V_square = length * width * height
When we multiply each dimension by the scale factor of 34, the new dimensions of the concert hall will be:
Length_concert_hall = 34 * length_model
Width_concert_hall = 34 * width_model
Height_concert_hall = 34 * height_model
Using these new dimensions, we can calculate the volume of the actual concert hall, V_concert_hall:
V_concert_hall = Length_concert_hall * Width_concert_hall * Height_concert_hall
So, the volume of the actual concert hall will be 34^3 times the volume of the model:
V_concert_hall = (34 * length_model) * (34 * width_model) * (34 * height_model)
Simplifying further, we get:
V_concert_hall = 34^3 * (length_model * width_model * height_model)
Therefore, the volume of the actual concert hall will be 34^3 times the volume of the square model.