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Sagot :
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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