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TMUA 2016 · Paper 2 · Question 19 of 20

TMUA 2016 Paper 2 Question 19

Coordinate geometry — Plans and elevations · reconstructing a solid from three views. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 2Coordinate geometryPlans and elevations · reconstructing a solid from three views7 options

The original question includes a diagram: Three 3-by-3 grids of squares: a staircase for the side elevation, an L-shape with a notch for the front elevation, and an L-shaped footprint for the plan view.

Some identical unit cubes are used to construct a three-dimensional object by gluing them together face to face.

Sketches of this object are made by looking at it from the right-hand side, from the front and from above. These sketches are called the side elevation, the front elevation, and the plan view respectively. Each sketch is drawn on a 3 by 3 grid of unit squares.

Side elevation: the whole bottom row (3 squares), the left two squares of the middle row, and the middle square of the top row.

Front elevation: the whole bottom row (3 squares), the left and right squares of the middle row (with a gap in the middle), and the right-hand square of the top row.

Plan view: the whole bottom row (3 squares) and the right-hand square of each of the other two rows, forming an L shape.

How many cubes were used to construct the object?

  1. Aexactly 6
  2. Beither 6 or 7
  3. Cexactly 7
  4. Deither 7 or 8
  5. Eexactly 8
  6. Feither 8 or 9
  7. Gexactly 9
Show the answer and worked solution
answer · F
  1. Aexactly 6
  2. Beither 6 or 7
  3. Cexactly 7
  4. Deither 7 or 8
  5. Eexactly 8
  6. Feither 8 or 9
  7. Gexactly 9
The plan view gives the footprint: five columns of cubes, three in a row across the front and two more running back from the right-hand end. Label the front row of columns 1, 2, 3 from left to right. The front elevation shows heights 2, 1 and 3 above positions 1, 2, 3, and since only one column stands at each of positions 1 and 2, those columns have heights 2 and 1. The side elevation shows the greatest height at each depth: 2 at the front, 3 in the middle, 1 at the back, so the two back columns have heights 3 and 1. That fixes four columns totalling 7 cubes. The remaining column, at the front right, only has to be no taller than 2 — both 1 and 2 are consistent with all three views — so the total is either 8 or 9.