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Calculating Volume of Right Cone Using Trigonometry

This guide explains how to find the volume of a right cone by applying trigonometric principles. Using the example of a cone with a height of 16 units and a base measured indirectly through tangent functions, we illustrate the steps taken to find the radius. The volume is calculated using the formula V = (1/3)πr²h. The examples include finding the flow rate of sand through a funnel-shaped cone, showcasing the integration of geometry and real-world applications. The process includes solving composite solid volumes and using multi-step solutions for precision.

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Calculating Volume of Right Cone Using Trigonometry

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  1. Find the volume of the right cone. To find the radius r of the base, use trigonometry. opp. tan 65° = Adj. 16 tan 65° = r 16 r = tan 65° EXAMPLE 3 Use trigonometry to find the volume of a cone SOLUTION Write ratio. Substitute. ≈ 7.46 Solve for r.

  2. 1 1 V = (π r 2)h ≈ π(7.462)(16) ≈ 932.45 ft3 3 3 EXAMPLE 3 Use trigonometry to find the volume of a cone Use the formula for the volume of a cone.

  3. Find the volume of the solid shown. 1 1 = s3 + Bh 3 3 = 63 + (6)2 6 EXAMPLE 4 Find volume of a composite solid SOLUTION Write formulas. Substitute.

  4. ANSWER The volume of the solid is 288cubic meters. EXAMPLE 4 Find volume of a composite solid = 216 + 72 Simplify. = 288 Add.

  5. You are using the funnel shown to measure the coarseness of a particular type of sand. It takes 2.8 seconds for the sand to empty out of the funnel. Find the flow rate of the sand in milliliters per second. (1 mL = 1 cm3) EXAMPLE 5 Solve a multi-step problem SCIENCE

  6. V = (πr2)h 1 1 3 3 = π(42)(6) EXAMPLE 5 Solve a multi-step problem SOLUTION STEP 1 Find: the volume of the funnel using the formula for the volume of a cone. ≈ 101 cm3 = 101 mL

  7. 101 mL ≈ 36.07 mL/s 2.8 s ANSWER The flow rate of the sand is about 36.07milliliters per second. EXAMPLE 5 Solve a multi-step problem STEP 2 Divide: the volume of the funnel by the time it takes the sand to empty out of the funnel.

  8. 4. Find the volume of the cone at the right. Round your answer to two decimal places. opp. tan 40° = Adj. r tan 65° = 5.8 for Examples 3, 4 and 5 GUIDED PRACTICE SOLUTION To find the radius r of the base, use triangle. Write ratio. Substitute. 5.8 tan 40°= 4.87 Solve for r.

  9. 1 1 v = (π r 2)h ≈ π (4.87)2 (5.8) ≈ 143.92 in3 3 3 for Examples 3, 4 and 5 GUIDED PRACTICE Use the formula for the volume of a cone

  10. Volume of solid Volume of cylinder Volume of cone = + = π 32h + (πr2) h = π 32 10 + (π32) 5 1 1 3 3 for Examples 3, 4 and 5 GUIDED PRACTICE 5. A right cylinder with radius 3centimeters and height 10centimeters has a right cone on top of it with the same base and height 5centimeters. Find the volume of the solid. Round your answer to two decimal places. SOLUTION = 330 cm2

  11. What If?In Example 5, suppose a different type of sand is used that takes 3.2seconds to empty out of the funnel. Find its flow rate. 6. 101ml 31.56ml/s = 3.25 for Examples 3, 4 and 5 GUIDED PRACTICE SOLUTION Divide the volume of the funnel by the time it takes the sand to empty out of the funnel.

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