Alliance Form 3 P1 Q10 — Density, Mass & Volume of a Square Brass Plate

KCSE (Form 3) Form 3 Measurement

Published

The Question

“A square brass plate is 2 mm thick and has a mass of 1.05 kg. The density of brass is 8.4 grams per cubic centimetre. Find the length of the plate in centimetres.”

1

Picture the plate and write its volume

The plate is a square, so its length and width are equal, each labelled L. It is a thin slab, so its volume is length times width times thickness. Because length and width are both L, this becomes L squared multiplied by the thickness. Setting up this expression first tells us exactly what to do once we know the actual volume.

V=L×L×t=L2 tV = L \times L \times t = L^{2} \, t
2

Make the units match

The density is given in grams and centimetres, but the mass is in kilograms and the thickness is in millimetres. To use the density directly we must convert everything to grams and centimetres. Multiply the kilograms by 1000 to get grams, and divide the millimetres by 10 to get centimetres.

1.05 kg=1050 g1.05 \text{ kg} = 1050 \text{ g}
2 mm=0.2 cm2 \text{ mm} = 0.2 \text{ cm}
3

Find the volume from mass and density

Density is mass divided by volume, so rearranging gives volume as mass divided by density. Dividing the mass in grams by the density in grams per cubic centimetre leaves a volume in cubic centimetres, which is exactly the unit we want.

V=mρ=10508.4=125 cm3V = \frac{m}{\rho} = \frac{1050}{8.4} = 125 \text{ cm}^{3}
4

Solve for the length L

Now put the volume into the plate formula. Since the volume equals L squared times the thickness, divide the volume by the thickness to isolate L squared. Then take the square root to find L. The square root of 625 is 25, so the plate is 25 centimetres long.

L2 t=125⇒L2=1250.2=625L^{2} \, t = 125 \Rightarrow L^{2} = \frac{125}{0.2} = 625
L=625=25 cmL = \sqrt{625} = 25 \text{ cm}

Final Result

The length of the square brass plate is 25 centimetres.

Why this method works

Mass, volume and density are linked by a single relationship: density measures how much mass is packed into each unit of volume. Because the plate's mass and material are fixed, that relationship pins down its volume exactly, no matter its shape. The shape then converts that volume into a length: a thin square slab has volume equal to the square of its side times its thickness, so once the volume and thickness are known the side length is forced. The only real trap is units, since density mixes grams with centimetres while the data arrives in kilograms and millimetres, so converting first keeps every quantity consistent.

✓

Check by reversing: 25 x 25 x 0.2 = 125 cubic cm, and 125 x 8.4 = 1050 g = 1.05 kg, matching the given mass.