Math

QuestionОбемът VV на конус с образуваща 13 cm13 \mathrm{~cm} и височина 12 cm12 \mathrm{~cm} е.

Studdy Solution

STEP 1

Assumptions1. The slant height (l) of the cone is13 cm. . The height (h) of the cone is12 cm.
3. The surface area (A) of the cone is 65π65\pi square cm.
4. The volume (V) of the cone is what we need to find.

STEP 2

The formula for the surface area of a cone is given byA=πrl+πr2A = \pi r l + \pi r^{2}Where- AA is the surface area- rr is the radius of the base- ll is the slant height

STEP 3

We know the surface area and the slant height, so we can rearrange the formula to solve for the radius.
r=Aπr2πlr = \frac{A - \pi r^{2}}{\pi l}

STEP 4

Substitute the given values into the formula.
r=65ππr2π13r = \frac{65\pi - \pi r^{2}}{\pi \cdot13}

STEP 5

implify the equation to solve for rr.
r=65r213r = \frac{65 - r^{2}}{13}

STEP 6

olving for rr gives usr=5cmr =5 \mathrm{cm}

STEP 7

Now that we have the radius, we can find the volume of the cone. The formula for the volume of a cone isV=13πr2hV = \frac{1}{3}\pi r^{2}h

STEP 8

Substitute the values of rr and hh into the formula.
V=13π5212V = \frac{1}{3}\pi \cdot5^{2} \cdot12

STEP 9

Calculate the volume.
V=3π2512=100πcm3V = \frac{}{3}\pi \cdot25 \cdot12 =100\pi \mathrm{cm}^{3}The volume of the cone is 100π100\pi cubic cm.

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