Question

The approximate volume of a human eye is 6.5 cm³. The volume of a laboratory model (excluding base and stand) of the human eye is 1404 cm³.

1. State whether the scale factor k is less than, equals to or greater than 1. 
2. Calculate the: 

1. value of k 
2. diameter of the human eye if the radius of the model is 7.2 cm.
3. the external surface area of the human eye if the surface area of the model is 651.6 cm².

Answer 1

Given: Volume of a human eye = 6.5 cm³, volume of the laboratory model = 1404 cm³; model radius = 7.2 cm; surface area of the model = 651.6 cm².

Step-wise calculation:

1. Direction of scale factor k

If the model is larger than the real eye (model volume > real volume) then the linear scale k (model length : actual length) is greater than 1.

2. Find k from volumes

Volume scales as k³.

So, `k^3 = V_"model" /V_"actual"` 

= `1404/6.5`

Compute 1404 ÷ 6.5 = 216   ...(Since 6.5 × 216 = 1404)

Hence, k³ = 216

⇒ `k = root(3)(216) = 6`.

3. Diameter of the human eye given model radius 7.2 cm.

Linear scale relation: radius_model = k × radius_human.

So, `"radius"_"human" = "radius"_"model"/k`.

radius_human = 7.2 ÷ 6 = 1.2 cm.

Diameter_human = 2 × 1.2 = 2.4 cm.

4. External surface area of the human eye

Surface area scales as k².

So, `"Area"_"human" = "Area"_"model"/k^2`.

k² = 6² = 36.

Area_human = 651.6 ÷ 36

= 18.1 cm²