Height of the cone having volume as [tex]25.12 \text { cm}^{3}[/tex] and base area [tex]12.56 \text { cm}^{2}[/tex] is 6 cm.
Solution:
Given that volume of cone = [tex]25.12 \text{ cm}^{3}[/tex]
Area of base = [tex]12.56 \text { cm}^{2}[/tex]
Need to determine height of the cone.
Formula for volume of the cone is as follows
[tex]\mathrm{V}_{c}=\frac{1}{3} \pi r^{2}{h} \rightarrow (1)[/tex]
Area of circular base of cone = [tex]A_{b}=\pi r^{2}[/tex]
Replacing [tex]\pi r^{2} \text { by } A_{b}[/tex] in equation (1), we get
[tex]\mathrm{V}_{\mathrm{c}}=\frac{1}{3} \mathrm{A}_{\mathrm{b}} \mathrm{h}[/tex]
[tex]\Rightarrow \frac{3 \mathrm{V} c}{\mathrm{A}_{\mathrm{b}}}=h[/tex]
[tex]\Rightarrow h=\frac{3 \mathrm{v} c}{\mathrm{A}_{\mathrm{b}}} \rightarrow (2)[/tex]
In our case Volume of cone [tex]V_c= 21.12 \text{ cm}^3[/tex] and Area of base [tex]A_b=12.56 \text { cm}^2[/tex]
On substituting the values of volume and area in equation 2 we get
[tex]h=\frac{3 \times 25.12}{12.56}=6 \text{ cm }[/tex]
The height of the cone from the process of relating both the volume of the cone with the area base of the cone is 6cm.
The volume of the cone is the region enclosed by the cone and it is related to one-third of the circular base area of the cone.
To determine the height of the cone, we need to equate the formula used in determining the volume of the cone with the area of the base.
Using the formula for the volume of a cone we have;
[tex]\mathbf{V = \dfrac{1}{3} \pi r^2 h}[/tex]
The area of the base can be expressed as:
[tex]\mathbf{A_{base}= \pi r ^2 }[/tex]
Equating both equations from above together, we have:
[tex]\mathbf{V = \dfrac{1}{3}A_{base} h}[/tex]
Making height (h), the subject of the formula, we have:
[tex]\mathbf{h = \dfrac{3V}{A_{base}}}[/tex]
[tex]\mathbf{h = \dfrac{3\times 25.12 \ cm^3}{12.56 cm^2 }}[/tex]
h = 6 cm
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