Answer:
1) 5.5 inches
2) 263.1 square inches
Step-by-step explanation:
Question 1
A frustum of a cone is the solid shape formed by slicing a cone parallel to its base and removing the smaller cone from the larger one.
The slant height of a frustum of a cone is the length of the slanted side connecting the top and bottom edges of the frustum. So, in this case, the slant height is BC.
The radius of a circle is half its diameter. Given that the diameter of the smaller base is 4 in, then its radius AB is 2 in.
Given that the diameter of the larger base is three times the diameter of the smaller base, then it measures 12 in. As OC is the radius of the larger base, then OC = 6 in.
Triangle BDC in the given diagram is a right triangle, where its base DC is the difference between the radii of the two bases of the frustrum. Therefore:
DC = OC - AB
DC = 6 - 2
DC = 4 in
To find the length of BC (the hypotenuse of right triangle BDC), we can use the cosine trigonometric ratio, which is defined as the ratio of the length of the adjacent side to the length of the hypotenuse:
[tex]\cos BCD = \dfrac{DC}{BC}[/tex]
Given that the angle between a slant height (BC) and a radius of the larger base (OC) is 43°, then m∠BCD = 43°. Therefore:
[tex]\cos 43^{\circ} = \dfrac{4}{BC}\\\\\\BC = \dfrac{4}{\cos 43^{\circ}}\\\\\\BC=5.46930984439...\\\\\\BC=5.5\; \sf in\; (nearest\;tenth)[/tex]
So, the length of the slant height is 5.5 in.
[tex]\dotfill[/tex]
Question 2
The formula for the total surface area of the frustum of a cone is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Surface Area of the Frustum of a Cone}}\\\\SA=\pi (r^2+R^2) + \pi l (r+R)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$SA$ is the surface area.}\\\phantom{ww}\bullet\;\textsf{$r$ and $R$ are the radii of the two circular bases.}\\\phantom{ww}\bullet\;\textsf{$l$ is the slant height.}\end{array}}[/tex]
In this case:
[tex]r = AB = 2[/tex]
[tex]R = OC = 6[/tex]
[tex]l = BC = \dfrac{4}{\cos 43^{\circ}}[/tex]
Substitute the values into the formula:
[tex]SA=\pi (2^2+6^2) + \pi\cdot \dfrac{4}{\cos 43^{\circ}} (2+6)[/tex]
Solve for SA:
[tex]SA=\pi (4+36) + \pi\cdot \dfrac{4}{\cos 43^{\circ}} (8)\\\\\\\\SA=40\pi+\dfrac{32\pi}{\cos 43^{\circ}}\\\\\\\\SA=263.122455162...\\\\\\SA=263.1\; \sf in^2\;(nearest\;tenth)[/tex]
[tex]\dotfill[/tex]
Additional Notes
Unless specified, we should always use exact values when calculating and avoid rounding mid-calculation. Therefore, the surface area of the frustrum has been calculated using the exact length of the slant height and the unrounded value of π, as question 2 did not specify using the rounded slant height or π ≈ 3.14.
