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The function h(t) below gives the approximate height of the water at Howth Harbour on a particular day, from 12 noon to 5 p.m - Junior Cycle Mathematics - Question 14 - 2016
Question 14
The function h(t) below gives the approximate height of the water at Howth Harbour on a particular day, from 12 noon to 5 p.m. h(t) = 10t² - 50t + 130, where h(t) ... show full transcript
Worked Solution & Example Answer:The function h(t) below gives the approximate height of the water at Howth Harbour on a particular day, from 12 noon to 5 p.m - Junior Cycle Mathematics - Question 14 - 2016
Step 1
Draw the graph of the function h(t) = 10t² - 50t + 130
Answer
- Calculate the vertex of the parabola to find the highest or lowest point.
- Determine the x-intercepts by solving the equation 10t² - 50t + 130 = 0.
- Plot the vertex and x-intercepts on the graph.
- Create a smooth curve passing through the calculated points to complete the graph.
The graph should have a vertex and a curve shaped like a parabola, indicating the height of the water over time from 12 noon to 5 p.m.
Step 2
Step 3
Estimate the height of the water at its lowest point.
Answer
The lowest point occurs at the vertex of the parabola, found at t = 2.5 hours (using vertex formula t = -b/(2a)).
Calculating:
h(2.5) = 10(2.5)² - 50(2.5) + 130 = 67.5 ext{ cm}.
Thus, the estimated height of the water at its lowest point is 67.5 cm.
Step 4
Step 5
Step 6
Hence, or otherwise, find the value of a and the value of b.
Answer
Using the second point (3, 180):
g(3) = a(3)² + b(3) + c = 180.
Substituting c = 180 gives:
9a + 3b + 180 = 180
=> 9a + 3b = 0
=> 3a + b = 0
=> b = -3a.
Using the lowest point (3, 0):
g(3) = a(3)² + b(3) + 180 = 0
=> 9a + 3b + 180 = 0.
Substituting b:
9a + 3(-3a) + 180 = 0
=> 9a - 9a + 180 = 0
=> a = 20, b = -60.
Thus, a = 20 and b = -60.