Surface Area of Cone

Which of the following objects is most likely to have a conical surface - a ball, a funnel, a plate, or a bed?

Solution:

From the list of items, only a funnel has a conical surface.

Find the curved surface area of a cone with radius \(7\, cm\) and slant height \(10\, cm\). Take \(\pi=\frac{22}{7}\)

Solution:

Since pi, radius, and slant height have been given, you should apply the formula. Hence the curved surface area of the cone is calculated as

\[A_{cs}=\frac{22}{7}\times 7\, cm \times 10\, cm\]

\[A_{cs}=220\, cm^2\]

Given a cone with a base radius of 7 feet and an internal height of 12 feet, calculate the surface area.

Solution:

As we have been given the internal height, we need to use Pythagoras' theorem to calculate the slant height:

7² + 12² = 193

Slant height =$\sqrt{193}$

We can take the formula and see what numbers we can plug into it: $a={\mathrm{\pi r}}^2+\mathrm{\pi rl}$

7 is our radius r, and $\sqrt{193}$ is our slant height l.

$\Rightarrow a=(\mathrm{\pi }\times 7^2)+(\mathrm{\pi }\times 7\times \sqrt{193})$

$\Rightarrow a=49\mathrm{\pi }+305.511$

$\Rightarrow a=459.45$

So our final answer, in this case, would be that$a=459.45f{t}^2$, as the area is measured in units².

Given a cone with a base diameter of 14 feet and an internal height of 18 feet, calculate the surface area.

Solution:

We need to be careful in this case, as we've been given the bottom length as diameter and not a radius. The radius is simply half of the diameter, so the radius in this case is 7 feet. Again, we need to use the Pythagorean theorem to calculate the slant height:

${18}^2+7^2=373$

Slant height = $\sqrt{373}$

We take the formula and then substitute r for 7 and l for $\sqrt{373}$:

$\Rightarrow a=(\mathrm{\pi }\times 7^2)+(\mathrm{\pi }\times 7\times \sqrt{373})$

$\Rightarrow a=49\mathrm{\pi }+424.720$

$\Rightarrow a=578.66$

Therefore, our final answer is $a=578.66f{t}^2$

From the figure below find the curved surface area of the cone.

Examples of curved surface are without the slant height, StudySmarter Originals

Take \(\pi=3.14\)

Solution:

In this problem, you have been given the radius and height but not the slant height.

By using Pythagoras theorem,

\[l=\sqrt{8^2+3.5^2}\]

\[l=8.73\, m\]

Now you can find the curved surface area

\[A_{cs}=3.14\times 3.5\, m \times 8.73\, m\]

Thus, the curved surface area of the cone, \(A_{cs}\) is:

\[A_{cs}=95.94\, m^2\]

In Ikeduru palm fruits are arranged in a conical manner, they are required to be covered with palm fronds of average area \(6\, m^2\) and mass \(10\, kg\). If the palm is inclined at an angle \(30°\) to the horizontal, and the base distance of a conical stock of palm fruits is \(100\, m\). Find the mass of palm frond needed to cover the stock of palm fruits. Take \(\pi=3.14\).

Solution:

Make a sketch of the story.

Finding the area of a cone with a given angle, StudySmarter Originals

So you can use SOHCAHTOA to get your slant height since

\[\cos\theta=\frac{adjacent}{hypotenuse}\]

\[\cos(30°)=\frac{50\, m}{l}\]

Cross multiply

\[0.866l=50\, m\]

Divide both sides by \(0.866\) to get the slant height, \(l\)

\[l=57.74\, m\]

Now you can find the total surface area of the conical stock knowing that

\[a=\pi r^2+\pi rl\]

Hence

\[a=(3.14\times (50\, m)^2)+(3.14\times 50\, m \times 57.74\, m)\]

\[a=7850\, m^2+9065.18\, m^2\]

Hence, the area of the conical stock is \(16915.18\, m^2\).

However, your task is to know the weight of palm fronds used to cover the conical stock. To do this, you need to know how many palm fronds would cover the stock since the area of a palm frond is \(6\, m^2\). Thus the number of palm fronds required, \(N_{pf}\) is

\[N_{pf}=\frac{16915.18\, m^2}{6\, m^2}\]

\[N_{pf}=2819.2\, fronds\]

With each palm frond weighing \(10\, kg\), the total mass of frond needed to cover the conical palm fruit stock, \(M_{pf}\) is:

\[M_{pf}=2819.2 \times 10\, kg\]

\[M_{pf}=28192\, kg\]

Therefore the mass of palm frond required to cover an average conical stock of palm fruit in Ikeduru is \(28192\, kg\).