I have two mathematically similar shapes, one of which has a volume of 1 litre and one which has a volume of half a litre. If the largest shape has a height of 40 cm, what (in centimetres) is the height of the smaller shape?
The volumes of similar shapes are proportional to the cubes of their corresponding sides, so
1 L / .5 L = 40^3 cm^3 / s^3 cm^3
cancelling the units
1 / .5 = 40^3/s^3
s^3 = .5(64,000) = 32,000
s = 32000^(1/3) = appr 31.748 cm
check"
40^3 : 31.748^3
= 64000 : 32000
= 2 : 1
the ratio of volumes is the cube of the ratio of heights.
So the ratio of heights is 1:∛2
40/∛2 = 31.748
To find the height of the smaller shape, we can use the concept of similar shapes and the ratio of their volumes.
Let's assume the height of the smaller shape is "h."
Given that the volume of the larger shape is 1 litre and the volume of the smaller shape is half a litre, we can set up the following proportion:
(volume of larger shape) / (volume of smaller shape) = (height of larger shape) / (height of smaller shape)
1 litre / 0.5 litres = 40 cm / h
Now we can solve for "h" by cross multiplying:
1 litre * h = 0.5 litres * 40 cm
h = (0.5 litres * 40 cm) / 1 litre
To simplify the calculation, we can convert litres to cm^3:
1 litre = 1000 cm^3
h = (0.5 * 40 * 1000) / 1
h = 20,000 / 1
h = 20,000 cm
Therefore, the height of the smaller shape is 20,000 centimeters.
To find the height of the smaller shape, we need to understand the concept of similarity in mathematics. When two shapes are mathematically similar, it means they have the same shape but different sizes.
In this case, we have two mathematically similar shapes with volumes of 1 litre and half a litre. Since the volumes of similar shapes are related by a scaling factor, we can set up a proportion to solve the problem.
First, let's denote the height of the smaller shape as "h" (in centimeters). We know that the largest shape has a height of 40 cm. So, using the volume ratio, we can set up the proportion:
(height of larger shape) / (height of smaller shape) = (volume of larger shape) / (volume of smaller shape)
Substituting the given values:
40 cm / h = 1 L / 0.5 L
To simplify the equation, we can convert the volumes from liters to centimeters cubed. Since 1 liter is equal to 1000 cm³:
40 cm / h = 1000 cm³ / 500 cm³
Now, cross-multiply and solve for "h":
40 * 500 = h * 1000
20,000 = h * 1000
Dividing both sides by 1000:
20,000 / 1000 = h
20 = h
Therefore, the height of the smaller shape is 20 centimeters.