Question

A marble statue has a mass of 1600 kg and is
270 cm tall.
The density of marble is 2500 kg/m³.
Justin makes a mathematically similar model
of the statue out of clay.
The model is 45 cm tall and has a density of
1200 kg/m³.
What is the mass of Justin's model?
Give your answer to 3 significant figures

Answer 1

Answer:

  3.56 kg

Step-by-step explanation:

You want the mass of a model that is 45 cm tall and has a density of 1200 kg/m³ when the statue it is modeling is 270 cm tall, has a density of 2500 kg/m³, and a mass of 1600 kg.

Volume

The ratio of volumes of the model to the statue is the cube of the ratio of their heights:

  Vm/Vs = (Hm/Hs)³

  Vm = Vs(Hm/Hs)³ = (1600 kg)/(2500 kg/m³)·((45 cm)/(270 cm))³

  Vm ≈ 0.002963 m³

Mass

The mass of the model is the product of its volume and its density:

  Mm = Vm·ρ = (0.002963 m³)(1200 kg/m³) ≈ 3.56 kg

The mass of Justin's model is about 3.56 kg.

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Additional comment

The relationship between density, volume, and mass is ...

  ρ = mass/volume

This can be rearranged to ...

  volume = mass/ρ

Which is the expression we used for Vs in the first section above.

(We used V and H for volume and height with 'm' and 's' signifying the model and the statue, respectively.)

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Answer 2

The mass of Justin's model is approximately 3.556 kg, rounded to 3 significant figures.

How to find the mass of Justin's model

Using the concept of Geometric similarity states that if two objects are mathematically similar, the ratio of their corresponding dimensions  is equal to the ratio of their corresponding volumes.

In this case, the ratio of the height of the statue to the height of the model is:

270 cm / 45 cm = 6

Since the model is mathematically similar to the statue, the ratio of their volumes will be the cube of the ratio of their heights:

$(6)^3$= 216

The volume of the statue can be calculated using its mass and density:

Volume of the statue = Mass of the statue / Density of marble

Volume of the statue = 1600 kg / 2500 kg/m³ = 0.64 m³

Now, we can calculate the volume of Justin's model:

Volume of the model = Volume of the statue / (Ratio of volumes)

Volume of the model = 0.64 m³ / 216 = 0.002963 m³

Finally, we can find the mass of Justin's model using its volume and density:

Mass of the model = Volume of the model * Density of clay

Mass of the model = 0.002963 m³ * 1200 kg/m³ = 3.556 kg

Therefore, the mass of Justin's model is approximately 3.556 kg, rounded to 3 significant figures.

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