Answer:
7.4 ft³
Step-by-step explanation:
To answer this question, we must assume that the circular bases of the cylinder touch all four sides of the bases of the prism, implying that the bases of the rectangular prism are squares. (If this is not the case, the question is unsolvable without additional information).
The formula to find the side length (s) of a square given its diagonal (d) is:
[tex]s=\sqrt{\dfrac{d^2}{2}}[/tex]
Given that the diagonals of the square bases are 2 ft long, then the side length of the bases of the prism is:
[tex]s=\sqrt{\dfrac{2^2}{2}}\\\\\\s=\sqrt{2}\; \sf ft[/tex]
If the bases of the rectangular prism are squares, then the radius of the circular bases of the cylinder is equal to half the side length of the square base. Therefore, the radius (r) of the cylinder is:
[tex]r=\dfrac{\sqrt{2}}{2}\sf \; ft[/tex]
The segment connecting vertex A of the top base of the prism with the centerpoint O of the bottom base of the cylinder makes an angle of 78° with the bottom base. Therefore, the measure of angle AOB in right triangle AOB is 78°. As the base of this triangle (OB) is half the length of the diagonal of the base, then OB = 1. To find the height of the triangle (AB), which is also the height of the cylinder, we can use the tangent trigonometric ratio:
[tex]\tan AOB=\dfrac{AB}{OB}\\\\\\\tan 78^{\circ}=\dfrac{AB}{1}\\\\\\AB=\tan 78^{\circ}[/tex]
Now, substitute the radius (r) and height (AB) into the volume of a cylinder formula to find the volume of the removable cylindrical bucket:
[tex]V=\pi r^2 h\\\\\\V=\pi \cdot OE^2 \cdot AB\\\\\\V=\pi \cdot \left(\dfrac{\sqrt{2}}{2}\right)^2\cdot \tan 78^{\circ}\\\\\\V=\pi \cdot \dfrac{1}{2}\cdot \tan 78^{\circ}\\\\\\V=7.390015694...\\\\\\V=7.4\; \sf ft^3[/tex]
Therefore, the volume of the removable cylindrical bucket rounded to the nearest tenth is 7.4 ft³.
