Question

finding missing height
notes:
the oblique cylinder has a radius of 3. side ab is 6.4 units long and creates an angle of 80 degrees with the base of the cylinder.

1. find the height of the cylinder using trig functions
2. find the volume of the cylinder

you try!
find the volume of the following 3d shapes.

1. volume = ________
2. volume = ________
3. volume = ________

Explanation:

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Part 1: Oblique Cylinder (Notes Section)

Step1: Find height via cosine

The height $h$ is adjacent to the 80° angle in a right triangle with hypotenuse $AB=6.4$.
$\cos(80^\circ) = \frac{h}{6.4}$
$h = 6.4 \times \cos(80^\circ)$
$h \approx 6.4 \times 0.1736 \approx 1.111$ units

Step2: Calculate base area

Base is a circle with radius $r=3$.
$A = \pi r^2 = \pi \times 3^2 = 9\pi$

Step3: Find cylinder volume

Volume of oblique cylinder = base area × height.
$V = 9\pi \times 1.111 \approx 31.1$ cubic units
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Part 2: You Try!

1. Oblique Prism

Step1: Find prism height via sine

Height $h$ is opposite 60° in triangle with hypotenuse 3m.
$\sin(60^\circ) = \frac{h}{3}$
$h = 3 \times \frac{\sqrt{3}}{2} \approx 2.598$ m

Step2: Calculate base area

Base is a rectangle: $A = 12 \times 2.5 = 30$ m²

Step3: Find prism volume

Volume = base area × height.
$V = 30 \times 2.598 \approx 77.9$ m³

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2. Oblique Cylinder

Step1: Find radius from diameter

Diameter $D=4$ cm, so radius $r = \frac{4}{2} = 2$ cm

Step2: Find cylinder height via sine

Height $h$ is opposite 70° in triangle with hypotenuse 6cm.
$\sin(70^\circ) = \frac{h}{6}$
$h = 6 \times \sin(70^\circ) \approx 6 \times 0.9397 \approx 5.638$ cm

Step3: Calculate base area

$A = \pi r^2 = \pi \times 2^2 = 4\pi$

Step4: Find cylinder volume

$V = 4\pi \times 5.638 \approx 70.8$ cm³

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3. Cone

Step1: Calculate base area

Base is a circle with radius $r=6$ cm.
$A = \pi r^2 = \pi \times 6^2 = 36\pi$

Step2: Find cone volume

Volume formula: $V = \frac{1}{3} \times$ base area × height.
$V = \frac{1}{3} \times 36\pi \times 16 = 192\pi \approx 603.2$ cm³

Answer:

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Notes Section:

1. Height of cylinder: $\approx 1.11$ units
2. Volume of cylinder: $\approx 31.1$ cubic units

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You Try!

1. Volume = $\approx 77.9$ m³
2. Volume = $\approx 70.8$ cm³
3. Volume = $\approx 603.2$ cm³ (or $192\pi$ cm³)