Question - Geometry Table Completions
Solution:
To complete the table, we will use the following relationships and formulas:1. The diameter of a circle is twice the radius: $$d = 2r$$.2. The area of the base (which is a circle) is given by $$A = \pi r^2$$.3. The volume of a cone is given by $$V = \frac{1}{3} \pi r^2 h$$, where $$h$$ is the height of the cone.Let's fill in the missing values for each row of the table:For the second row:- Given the diameter as 3 units, the radius $$r$$ is half of that, so $$r = \frac{3}{2}$$ units.- The area of the base $$A$$ is $$\pi r^2 = \pi (\frac{3}{2})^2 = \frac{9}{4} \pi$$ square units.- The given volume $$V$$ is $$27\pi$$ cubic units, we can use it to find the height $$h$$: $$V = \frac{1}{3} \pi r^2 h$$ $$27\pi = \frac{1}{3} \pi \left(\frac{3}{2}\right)^2 h$$ $$27\pi = \frac{1}{3} \pi \frac{9}{4} h$$ $$27\pi = \pi \frac{9}{4} \frac{1}{3} h$$ $$27 = \frac{9}{12} h$$ $$h = 27 \times \frac{12}{9}$$ $$h = 36$$ So the height is 36 units.For the third row:- Given the radius $$r = 10$$ units, the diameter $$d$$ is twice that, so $$d = 20$$ units.- The area of the base $$A$$ is $$\pi r^2 = \pi (10)^2 = 100\pi$$ square units.- The height $$h$$ is 12 units, as given, so no calculation is needed for that.For the fourth row:- Given the volume $$V = 3.14$$ cubic units and height $$h = 3$$ units, we can find the radius $$r$$: $$V = \frac{1}{3} \pi r^2 h$$ $$3.14 = \frac{1}{3} \pi r^2 \cdot 3$$ $$3.14 = \pi r^2$$ $$r^2 = \frac{3.14}{\pi}$$ Because $$\pi$$ is approximately 3.14, $$r^2$$ is approximately 1, hence $$r \approx 1$$ unit.- The approximate diameter $$d$$ is twice the radius, so $$d \approx 2$$ units.- The area of the base $$A$$ is $$\pi r^2 = \pi (1)^2 = \pi$$ square units, which we can approximate as 3.14 square units since the volume was given to this level of precision.The completed table should look like this:- Diameter: 2, 3, 20, 2- Radius: 1, 1.5, 10, 1- Area of the base: $$\pi$$, $$2.25\pi$$, $$100\pi$$, $$\pi$$- Height: 3, 36, 12, 3- Volume of Cone: $$3.14$$, $$27\pi$$, $$400\pi$$, $$3.14$$
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