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Question 1) A sculptor created a marble water basin by beginning with a cylinder and then carving out a hemisphere. The cylinder has a radius of 30 centimeters and a height of 45 centimeters. The hemisphere has a radius of 25 centimeters.

What is the approximate volume of the marble that forms the water basin?

A) 123,307.5 cubic centimeters
B) 61,784.7 cubic centimeters
C) 125,925.5 cubic centimeters
D) 94,509.6 cubic centimeters

Question 2) Toby made a heart-shaped cake for his mom's birthday. He baked a square cake and a round cake. One side of the square cake is 9 inches long, and the diameter of the round cake is 9 inches. Both cakes have a height of 3 inches. He cut the round cake in half and attached it to two sides of the square cake to form the heart. He covered the combined cake in red frosting before adding decorations.

What is the approximate area that Toby covered in red frosting?

A) 364 square inches
B) 283 square inches
C) 302 square inches
D) 198 square inches

Question 3)
A designer is creating a window for a new home. The window is a semicircle with a radius of 9 inches placed at the top edge of an 18-inch by 24-inch rectangle. She plans to put a wooden frame around the outside of the glass.

What is the approximate length of the frame around the window?

A) 122.5 inches
B) 94.3 inches
C) 112.3 inches
D) 140.5 inches

Sagot :

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1) The sculptor created a marble basin with an approximate volume of 94,509.6 cubic centimeters -  Option D, 2) The approximate area of the heart-shaped cake is 283 square inches - Option B, 3) The length of the wooden frame around the window is 94.3 inches - Option B.

In this question we should make use of geometric formulas for volumes and areas and key information from statement in order to find the right choices.

1) In this case, the volume occupied by the marble water basin ([tex]V[/tex]), in cubic centimeters, by subtracting the volume of the hemisphere from the volume of the cylinder:

[tex]V = \pi\cdot R^{2}\cdot h - \frac{2\pi}{3} \cdot r^{3}[/tex] (1)

Where:

  • [tex]R[/tex] - Radius of the cylinder, in centimeters.
  • [tex]h[/tex] - Height of the cylinder, in centimeters.
  • [tex]r[/tex] - Radius of the cylinder, in centimeters.

If we know that [tex]R = 30\,cm[/tex], [tex]h = 45\,cm[/tex] and [tex]r = 25\,cm[/tex], then the volume of the marble is:

[tex]V = \pi \cdot (30\,cm)^{2}\cdot (45\,cm) - \frac{2\pi}{3}\cdot (25\,cm)^{3}[/tex]

[tex]V \approx 94509.579\,cm^{3}[/tex]

The right choice is D.

2) To determine the approximate surface area of the cake covered in red frosting ([tex]A_{s}[/tex]), in square inches, we need to find the sum of the surface area of the entire circle and the surface area of the square:

[tex]A_{s} = l^{2} + 2\cdot l\cdot h + \frac{\pi}{4}\cdot D^2 + \pi \cdot D\cdot h[/tex] (2)

Where:

  • [tex]l[/tex] - Side length of the square, in inches.
  • [tex]h[/tex] - Height of the cakes, in inches.
  • [tex]D[/tex] - Diameter of the cakes, in inches.

If we know that [tex]l = 9\,in[/tex], [tex]h = 3\,in[/tex] and [tex]D = 9\,in[/tex], then the approximate area covered in red frosting:

[tex]A_{s} = (9\,in)^{2} + 2\cdot (9\,in)\cdot (3\,in) + \frac{\pi}{4}\cdot (9\,in)^{2} + \pi \cdot (9\,in)\cdot (3\,in)[/tex]

[tex]A_{s} \approx 283.440\,in^{2}[/tex]

The right choice is B.

3) The frame around the window is found by means of the following perimeter formula ([tex]s[/tex]), in inches:

[tex]s = \pi\cdot r + 2\cdot h + 2\cdot r[/tex] (3)

Where:

  • [tex]r[/tex] - Radius, in inches.
  • [tex]h[/tex] - Height of the rectangle, in inches.

If we know that [tex]r = 9\,in[/tex] and [tex]h = 24\,in[/tex], then the length of the frame around the window is:

[tex]s = \pi\cdot (9\,in) + 2\cdot (24\,in) + 2\cdot (9\,in)[/tex]

[tex]s \approx 94.274\,in[/tex]

The right choice is B.

We kindly invite to see question on volumes: https://brainly.com/question/1578538

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