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

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

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

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

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

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. 101 mL 2.8 s ≈ 36.07 mL/s The flow rate of the sand is about 36.07 milliliters per second. ANSWER

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

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

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

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