Question

volume
figure
surface area

1. calculate the volume and surface area for the cone with diameter 18 ft and height 27 ft
2. calculate the volume and surface area for the cone with height 15 cm and diameter 24.6 cm
3. calculate the volume and surface area for the cone with radius 11.5 m and height 25 m

Explanation:

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Cone 1 (18 ft diameter, 27 ft height)

Step1: Find base radius

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

Step2: Calculate volume

Volume formula: $V = \frac{1}{3}\pi r^2 h$
$V = \frac{1}{3}\pi (9)^2 (27) = \frac{1}{3}\pi (81)(27) = 729\pi$ cubic ft

Step3: Calculate slant height $l$

Use Pythagorean theorem: $l = \sqrt{r^2 + h^2}$
$l = \sqrt{9^2 + 27^2} = \sqrt{81 + 729} = \sqrt{810} = 9\sqrt{10}$ ft

Step4: Calculate surface area

Surface area formula: $SA = \pi r(r + l)$
$SA = \pi (9)(9 + 9\sqrt{10}) = 81\pi(1 + \sqrt{10})$ square ft

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Cone 2 (24.6 cm diameter, 15 cm height)

Step1: Find base radius

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

Step2: Calculate volume

$V = \frac{1}{3}\pi (12.3)^2 (15) = \frac{1}{3}\pi (151.29)(15) = 756.45\pi$ cubic cm

Step3: Calculate slant height $l$

$l = \sqrt{12.3^2 + 15^2} = \sqrt{151.29 + 225} = \sqrt{376.29} \approx 19.4$ cm

Step4: Calculate surface area

$SA = \pi (12.3)(12.3 + 19.4) = 12.3\pi(31.7) = 389.91\pi$ square cm

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Cone 3 (11.5 m radius, 25 m height)

Step1: Calculate volume

$V = \frac{1}{3}\pi (11.5)^2 (25) = \frac{1}{3}\pi (132.25)(25) = \frac{3306.25}{3}\pi \approx 1102.08\pi$ cubic m

Step2: Calculate slant height $l$

$l = \sqrt{11.5^2 + 25^2} = \sqrt{132.25 + 625} = \sqrt{757.25} \approx 27.52$ m

Step3: Calculate surface area

$SA = \pi (11.5)(11.5 + 27.52) = 11.5\pi(39.02) = 448.73\pi$ square m

Answer:

1. Cone 1 (ft):

Volume: $729\pi$ cubic feet
Surface Area: $81\pi(1+\sqrt{10})$ square feet

1. Cone 2 (cm):

Volume: $756.45\pi$ cubic centimeters
Surface Area: $389.91\pi$ square centimeters

1. Cone 3 (m):

Volume: $\frac{3306.25}{3}\pi$ (or $\approx1102.08\pi$) cubic meters
Surface Area: $448.73\pi$ square meters

If numerical approximations are needed, use $\pi \approx 3.1416$ to calculate decimal values.