In a local ice sculpture contest, one group sculpted a block into a rectangular based pyramid. The dimensions of the base were 3

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In a local ice sculpture contest, one group sculpted a block into a rectangular based pyramid. The dimensions of the base were 3

m by 5 m, and the pyramid was 3.6 m high. Calculate the amount of ice needed for this sculpture. A conical-shaped umbrella has a radius of 0.4 m and a height of 0.45 m. Calculate the amount of fabric needed to manufacture this umbrella. (Hint: an umbrella will have no base) A cone has a volume of 150 cm3 and a base with an area of 12 cm2. What is the height of the cone? Find the dimensions of a deck which will have railings on only three sides. There is 28 m of railing available and the deck must be as large as possible. A winter recreational rental company is fencing in a new storage area. They have two options. They can set it up at the back corner of the property and fence it in on four sides. Or, they can attach it to the back of their building and fence it in on three sides. The rental company has decided that the storage area needs to be 100 m2 if it is in the back corner or 98 m2 if it is attached to the back of the building. Determine the optimal design for each situation.

Mathematics

1 answer:

Svet_ta [12.7K] · Answer 1 · 2026-04-09T07:09:43+03:00

Answer:

1. Total ice required = 18 m²

2. The fabric area necessary for the umbrella is 0.76 m²

3. The cone's height measures 3.75 cm

4. The deck's dimensions are as follows;

Width = 28/3 m, length = 28/3 m

The area amounts to 87.11 m²

5. When considering the optimal design for the storage area in the corner, we find;

Width = 10m

Length = 10 m

For the best configuration at the back of the building, the dimensions are;

Width = 7·√2 m

Length = 7·√2 m

Step-by-step explanation:

1. The required ice volume, V, is calculated using the formula, V = 1/3 × Base area × Height

The base area = Width × Length = 3 × 5 = 15 m²

The height of the pyramid is 3.6 m

Calculating the volume gives us V = 1/3*15*3.6 = 18 m²

Thus, the total ice required is 18 m²

2. The umbrella's surface area is the lateral area of a cone, excluding the base

The lateral area of a cone = π×r×l

Where:

r = Cone radius = 0.4 m

l = Slant height = √(h² + r²)

h = Cone height = 0.45 m

Calculating l gives: l = √(0.45² + 0.4²) = 0.6021 m

The surface area therefore equals π×0.4×0.6021 = 0.76 m²

Indicating the fabric area necessary for the umbrella is 0.76 m²

3. For the cone, the volume is given by V = 1/3×Base area, A, ×Height, h

Specifics provide V = 150 cm³

and A = 120 cm²

Thus, rearranging gives us;

h = 3×V/A = 3*150/120 = 3.75 cm

4. Regarding the deck, it will have railings on three sides, and dimensions should be calculated accordingly.

Max dimensions are aligned with a square for optimal area given by dimensions being equal, leading to the calculation of each side to be 28/3 m

Thus the deck dimensions of width = 28/3 m, length = 28/3 m provide an area of 28/3×28/3 = 784/9 = $87\frac{1}{9}$ =87.11 m²

5. The storage area at the corner of the property will have dimensions resulting in an optimal square of 10 m for each side;

Width = 10m

Length = 10 m

The back design with a fence on three sides should optimize the side length using s² = 98 m²

leading to s = √98 = 7·√2 m

Yielding dimensions of width = 7·√2 m and breadth = 7·√2 m.