Question

a canadian coast guard rescue boat is searching for a ship in distress. a spotter notices a flare that has been fired vertically from the ship in the distress. the height of the flare, h metres above the ocean, after t seconds is modelled by the quadratic function h(t)= - 4.9t²+167.8t. answer the following accurate to one decimal place. a) how high is the flare after 4 seconds? use a graphing calculator for these questions h=-4.9(4)²+167.8(4) h=-78.4 + 671.2 h = 592.8 m b) for how much time is the flare above 1 km? y₁=-4.9x²+167.8x y₂ = 1000 (1000m = 1km) window x(0,40) y(0,2000) 1st intersection point=(7.683348,1000) 2nd intersection point=(26.56155,1000) >1 km = 26.56155 - 7.683348 = 18.878202 ≈ 18.9 s c) how much time does the flare take to reach its maximum height?

Explanation:

Step1: Calculate height at \(t = 4\)

Substitute \(t=4\) into \(h(t)=-4.9t^{2}+167.8t\).
\[h=-4.9\times(4)^{2}+167.8\times(4)\]
\[h=-4.9\times16 + 671.2\]
\[h=-78.4+671.2\]
\[h = 592.8\]

Step2: Solve for \(t\) when \(h = 1000\) (since \(1\ km=1000\ m\))

Set \(h(t)=1000\), so \(-4.9t^{2}+167.8t = 1000\).
Rearrange to get \(4.9t^{2}-167.8t + 1000=0\).
Use the quadratic formula \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for \(at^{2}+bt + c = 0\). Here \(a = 4.9\), \(b=-167.8\), \(c = 1000\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-167.8)^{2}-4\times4.9\times1000\)
\[=28156.84 - 19600\]
\[=8556.84\]
Then \(t=\frac{167.8\pm\sqrt{8556.84}}{9.8}\)
\[t=\frac{167.8\pm92.5}{9.8}\]
We get two solutions:
\(t_1=\frac{167.8 + 92.5}{9.8}=\frac{260.3}{9.8}\approx26.56\)
\(t_2=\frac{167.8-92.5}{9.8}=\frac{75.3}{9.8}\approx7.68\)

Step3: Find time to reach maximum height

For a quadratic function \(y = ax^{2}+bx + c\), the time at which the maximum occurs is \(t=-\frac{b}{2a}\).
For \(h(t)=-4.9t^{2}+167.8t\), \(a=-4.9\), \(b = 167.8\).
\[t=-\frac{167.8}{2\times(-4.9)}=\frac{167.8}{9.8}\approx17.12\ s\]

Answer:

a) The height of the flare after 4 seconds is \(592.8\ m\).
b) The flare is above \(1\ km\) at approximately \(t = 7.68\ s\) and \(t = 26.56\ s\).
c) The flare takes approximately \(17.12\ s\) to reach its maximum height.