Question 21·25 Super-Hard SAT Geometry and Trigonometry Questions·Geometry and Trigonometry

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A decorative solid is formed by attaching the base of a right circular cone to the top of a right circular cylinder. Both solids have radius $r$ . The cylinder has height $2r$ , and the cone has vertical height $\frac{4r}{3}$ . The total volume of the solid is $1782\pi$ cubic units.

The exterior surface area includes the bottom circular base of the cylinder, the lateral surface of the cylinder, and the lateral surface of the cone. The circular face where the cone and cylinder meet is not exposed. For a cylinder, lateral surface area is $2\pi rh$ . For a cone, lateral surface area is $\pi r\ell$ , where $\ell$ is the slant height.

If the exterior surface area is $k\pi$ square units, what is the value of $k$ ?

Your answer

A composite solid is built from familiar shapes. Add the cylinder and cone volumes to find their shared radius, using a one-variable Desmos regression. Then use a right triangle for the cone’s slant height and add only the exposed surfaces. The shared circle is hidden, and using vertical height for the cone’s curved side gives too little area.

Hints

Hint 1
To find the radius, add the volumes of the attached solids. The cylinder uses $\pi r^2(2r)$ ; the cone uses one-third of $\pi r^2\left(\frac{4r}{3}\right)$ . What equation matches their total volume?
Hint 2
Both volume terms contain $\pi$ , so divide it out. In Desmos, a regression can find the radius: change $r$ to $r_1$ and $=$ to $\sim$ , then require a positive radius.
Hint 3
The cone’s slant height is the long side of a right triangle whose legs are the radius and vertical height. For exterior area, count the bottom and both curved sides, not the joined circle.

Step-by-step

Find the radius, then count exposed surfaces
Step 1
Write one equation for the total volume
The cone is attached, not cut out, so add its volume to the cylinder’s. Cylinder volume is $\pi r^2h$ ; cone volume is $\frac13\pi r^2h$ . Use each given vertical height and divide out the $\pi$ shared by every term:
$1782=r^2(2r)+\frac13r^2\left(\frac{4r}{3}\right).$
Step 2
Find the shared radius in Desmos
Type $r_1^2(2r_1)+\frac13r_1^2\left(\frac{4r_1}{3}\right)\sim1782\left\{r_1>0\right\}$ in Desmos. The subscript makes $r_1$ a value for the regression to find rather than a slider; the restriction keeps the radius positive. Desmos shows $r_1=9$ under PARAMETERS, so $r=9$ units.
Step 3
Find the cone’s slant height
From the center of the cone’s base to its edge is $r=9$ ; from that center straight up to the tip is $\frac{4(9)}3=12$ . Those lengths are perpendicular legs, and the slant height $\ell$ is the long side. The $9$ - $12$ - $15$ right-triangle triple gives $\ell=15$ , since $9^2+12^2=15^2$ . Don’t use $12$ for the curved surface: it’s the vertical height.
Step 4
Count only the outside surfaces
The bottom contributes $\pi r^2$ . The two lateral surfaces, or curved sides, contribute $2\pi r(2r)$ and $\pi r\ell$ . The cylinder’s top and cone’s base are hidden at the join, so neither counts:
$\text{exterior area}=\pi r^2+2\pi r(2r)+\pi r\ell.$
Step 5
Find the coefficient of pi
Enter $r_1^2+2r_1(2r_1)+r_1(15)$ in Desmos to add the three areas without their common $\pi$ . Desmos prints $540$ . The exterior surface area is $540\pi$ square units, so $k=540$ . Grid in 540.

Question 21·25 Super-Hard SAT Geometry and Trigonometry Questions·Geometry and Trigonometry

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