Nets, and views of a solid from the top, front and side

Answer Key - Grade 10 Math Worksheet - Challenge Me

Circle the best answer, or write your answer on the line.

1. Read this first

   Use this

   3 by 3 base of stacked cubes. Columns A, B, C run left to right as seen from the front; rows 1, 2, 3 run from the front to the back. Front view, tallest stack in each column: 3, 1, 1. Right-side view, tallest stack in each row (row 1 on the left): 2, 2, 3.

2. 1.

   Ari builds on a 3 by 3 base and records the front and right-side views on the card. What is the least number of cubes the building could have?

   9 cubes

3. Use this

   Columns A, B, C run left to right as seen from the front; rows 1, 2, 3 run from the front to the back. Front view, tallest stack in each column: 2, 3, 1. Right-side view, tallest stack in each row (row 1 on the left): 1, 1, 3.

4. 2.

   Bo builds on a 3 by 3 base and records the front and right-side views on the card. What is the least number of cubes the building could have?

   7 cubes
5. 3.

   Esme builds on a 3 by 3 base and records the front and right-side views on the card. What is the least number of cubes the building could have?

   13 cubes
6. 4.

   Jun makes stack A from 20 coins in a straight column, and stack B by sliding the same coins sideways into a leaning column of the same height. Which reply is justified?

   • Equal surface areas: slices of equal area give equal outsides
   • Equal volumes: slices at the same height have equal area (Cavalieri's principle)
   • The stacks are congruent solids, because each slice has the same area at each height, so they match exactly
   • Different volumes: the leaning stack is slanted, so it holds less
7. 5.

   Luz has a square prism with base area 36 cm² and a cylinder with base area 36 cm². Both are 5 cm tall. Which reply is justified?

   • Equal volumes, 180 cm³ each: slices at every height have equal area
   • Equal surface areas, since both bases are 36 cm²
   • The cylinder is larger, because a round base of area 36 cm² holds more than a square one
   • The solids are congruent, since their slices have equal area
8. 6.

   Milo drew two plans for one card. The card shows the front and right-side views. Which reply is justified?

   • Plan Y does not fit the card, because it holds 8 fewer cubes than plan X does
   • Both plans fit the card, with 14 and 6 cubes, so the views do not fix the building
   • Plan X does not fit the card, because 8 of its cubes are hidden and the card cannot show them
   • Both plans fit the card, so both buildings must hold 14 cubes, the count of plan X

9. Card

   Front view, tallest stack in each column from the left: 3, 3, 1; right-side view, tallest stack in each row (row 1 on the left): 1, 3, 2. Plan X, stack heights by row (columns left to right): row 1: 1, 1, 1; row 2: 3, 3, 1; row 3: 2, 2, 1. Plan Y: row 1: 0, 0, 1; row 2: 3, 3, 0; row 3: 0, 2, 0.

10. 7.

    Pia drew two plans for one card. The card shows the front and right-side views. Which reply is justified?

    • Both plans fit the card, so both buildings must hold 15 cubes, the count of plan X
    • Plan Y does not fit the card, because it holds 6 fewer cubes than plan X does
    • Both plans fit the card, with 15 and 9 cubes, so the views do not fix the building
    • Plan X does not fit the card, because 6 of its cubes are hidden and the card cannot show them
11. 8.

    Ren slices a cone with a plane through its apex and the center of its base. The slice is an isosceles triangle. Which reason proves it?

    • Two slant edges from the apex to the base circle are equal, and the base chord is a diameter
    • Slices of a cone are circles, so this triangle is really a circle seen edge-on, with two equal sides
    • The slant height equals the base radius, so the three sides of the slice are equal
    • The slice has one base and one height, so it must be a triangle with two equal sides
12. 9.

    Tala slices a sphere of radius 5 cm with a plane 3 cm from its center. How big is the circle, and why?

    • A circle with radius 5 cm, since a slice of the sphere has the sphere's radius
    • A circle with radius 3 cm, since the plane is 3 cm away from the center
    • A circle with radius 2 cm, since the radius is 5 − 3 = 2 cm
    • A circle with radius 4 cm, since 3² + 4² = 5²
13. 10.

    Zee compares a hemisphere of radius r with a cylinder (radius r, height r) minus a cone with its apex at the center. Which fact shows their slices at height d match?

    • The cone has a slice of area π(r² − d²) at height d, the same as the hemisphere
    • The cylinder minus the cone has a ring of area π(r − d)² at height d, the square of the difference
    • The cylinder minus the cone has a ring of area πr² − πd² at height d
    • The cylinder has a slice of area πr² at height d, so it matches the hemisphere