Question

volume
figure
surface area

1. a cone with a diameter of 18 ft and a height of 27 ft
2. a cone with a diameter of 24.6 cm and a height of 15 cm
3. a cone with a radius of 11.5 m and a height of 25 m

calculate the volume and surface area for each cone above.

Explanation:

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Cone 1 (18 ft diameter, 27 ft height)

Step1: Find radius of the cone

Diameter $d=18$ ft, so radius $r=\frac{d}{2}=\frac{18}{2}=9$ ft

Step2: Calculate volume of the cone

Volume formula: $V=\frac{1}{3}\pi r^2 h$
$V=\frac{1}{3} \times \pi \times 9^2 \times 27 = \frac{1}{3} \times \pi \times 81 \times 27 = 729\pi \approx 2289.06$ cubic feet

Step3: Calculate slant height $l$

Slant height formula: $l=\sqrt{r^2+h^2}$
$l=\sqrt{9^2+27^2}=\sqrt{81+729}=\sqrt{810}=9\sqrt{10} \approx 28.46$ ft

Step4: Calculate surface area of the cone

Surface Area formula: $SA=\pi r(r+l)$
$SA=\pi \times 9 \times (9+28.46)=9\pi \times 37.46 \approx 1356.48$ square feet

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Cone 2 (24.6 cm diameter, 15 cm height)

Step1: Find radius of the cone

Diameter $d=24.6$ cm, so radius $r=\frac{d}{2}=\frac{24.6}{2}=12.3$ cm

Step2: Calculate volume of the cone

Volume formula: $V=\frac{1}{3}\pi r^2 h$
$V=\frac{1}{3} \times \pi \times 12.3^2 \times 15 = \frac{1}{3} \times \pi \times 151.29 \times 15 = 756.45\pi \approx 2373.25$ cubic centimeters

Step3: Calculate slant height $l$

Slant height formula: $l=\sqrt{r^2+h^2}$
$l=\sqrt{12.3^2+15^2}=\sqrt{151.29+225}=\sqrt{376.29} \approx 19.40$ cm

Step4: Calculate surface area of the cone

Surface Area formula: $SA=\pi r(r+l)$
$SA=\pi \times 12.3 \times (12.3+19.40)=12.3\pi \times 31.70 \approx 902.63$ square centimeters

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Cone 3 (11.5 m radius, 25 m height)

Step1: Identify given values

Radius $r=11.5$ m, height $h=25$ m

Step2: Calculate volume of the cone

Volume formula: $V=\frac{1}{3}\pi r^2 h$
$V=\frac{1}{3} \times \pi \times 11.5^2 \times 25 = \frac{1}{3} \times \pi \times 132.25 \times 25 = \frac{3306.25}{3}\pi \approx 3406.03$ cubic meters

Step3: Calculate slant height $l$

Slant height formula: $l=\sqrt{r^2+h^2}$
$l=\sqrt{11.5^2+25^2}=\sqrt{132.25+625}=\sqrt{757.25} \approx 27.52$ m

Step4: Calculate surface area of the cone

Surface Area formula: $SA=\pi r(r+l)$
$SA=\pi \times 11.5 \times (11.5+27.52)=11.5\pi \times 39.02 \approx 1208.92$ square meters

Answer:

1. Volume: $2289.06$ cubic feet; Surface Area: $1356.48$ square feet
2. Volume: $2373.25$ cubic centimeters; Surface Area: $902.63$ square centimeters
3. Volume: $3406.03$ cubic meters; Surface Area: $1208.92$ square meters