I'll solve each problem step by step.
Problem 1:
Given: h(t) = -4.9t² + 9.8t + 28.224
When the ball strikes the ground, h(t) = 0
So: -4.9t² + 9.8t + 28.224 = 0
Using the quadratic formula: t = (-b ± (b² - 4ac)) / 2a
where a = -4.9, b = 9.8, c = 28.224
Discriminant: b² - 4ac = (9.8)² - 4(-4.9)(28.224)
= 96.04 + 553.1904
= 649.2304
√649.2304 ≈ 25.48
t = (-9.8 ± 25.48) / (2 × -4.9)
t = (-9.8 ± 25.48) / -9.8
Taking the positive root:
t = (-9.8 + 25.48) / -9.8 = 15.68 / -9.8 = -1.6 (reject, negative time)
Taking the other root:
t = (-9.8 - 25.48) / -9.8 = -35.28 / -9.8 = 3.6 seconds
Let me verify: h(3.6) = -4.9(3.6)² + 9.8(3.6) + 28.224
= -4.9(12.96) + 35.28 + 28.224
= -63.504 + 35.28 + 28.224
= -63.504 + 63.504 = 0 ✓
Answer for Problem 1: 3.6 seconds
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Problem 2:
Given: h(t) = -16t² + 65.6t + 480.32
When h(t) = 0: -16t² + 65.6t + 480.32 = 0
Using quadratic formula with a = -16, b = 65.6, c = 480.32
Discriminant: (65.6)² - 4(-16)(480.32)
= 4303.36 + 30740.48
= 35043.84
√35043.84 ≈ 187.2
t = (-65.6 ± 187.2) / (2 × -16)
t = (-65.6 ± 187.2) / -32
Positive solution:
t = (-65.6 - 187.2) / -32 = -252.8 / -32 = 7.9 seconds
Verify: h(7.9) = -16(7.9)² + 65.6(7.9) + 480.32
= -16(62.41) + 518.24 + 480.32
= -998.56 + 998.56 = 0 ✓
Answer for Problem 2: 7.9 seconds
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Problem 3:
Given: h(t) = -4.9t² + 4.41t + 53.998
When h(t) = 0: -4.9t² + 4.41t + 53.998 = 0
Using quadratic formula with a = -4.9, b = 4.41, c = 53.998
Discriminant: (4.41)² - 4(-4.9)(53.998)
= 19.4481 + 1058.3608
= 1077.8089
√1077.8089 ≈ 32.83
t = (-4.41 ± 32.83) / (2 × -4.9)
t = (-4.41 ± 32.83) / -9.8
Positive solution:
t = (-4.41 - 32.83) / -9.8 = -37.24 / -9.8 = 3.8 seconds
Verify: h(3.8) = -4.9(3.8)² + 4.41(3.8) + 53.998
= -4.9(14.44) + 16.758 + 53.998
= -70.756 + 70.756 = 0 ✓
Answer for Problem 3: 3.8 seconds
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Problem 4:
Given: h(t) = -16t² + 36.8t + 603.84
The initial height is when t = 0.
h(0) = -16(0)² + 36.8(0) + 603.84 = 603.84 feet
Answer for Problem 4: 603.84 feet
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Problem 5:
Given: h(t) = -4.9t² + 10.78t + 19.992
The initial height is when t = 0.
h(0) = -4.9(0)² + 10.78(0) + 19.992 = 19.992 meters
Answer for Problem 5: 19.992 meters
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Problem 6:
Given: h(t) = -16t² + 41.6t + 588
The initial height is when t = 0.
h(0) = -16(0)² + 41.6(0) + 588 = 588 feet
Answer for Problem 6: 588 feet
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Final Answer:
1. 3.6 seconds
2. 7.9 seconds
3. 3.8 seconds
4. 603.84 feet
5. 19.992 meters
6. 588 feet
Parent Tip: Review the logic above to help your child master the concept of quadratic formula word problems worksheets.
