11. [-/4 Points] DETAILS MY NOTES SCALCET9 2.7.012.MI. If a rock is thrown upward on the planet Mars with a velocity of \( 15 \mathrm{~m} / \mathrm{s} \), its height above the ground (in meters) a \( t \) seconds is given by \( \mathrm{H}=15 \mathrm{t}-1.86 \mathrm{t}^{2} \). (a) Find the velocity (in \( \mathrm{m} / \mathrm{s} \) ) of the rock after 2 seconds. (b) Find the velocity (in \( \mathrm{m} / \mathrm{s} \) ) of the rock when \( t=a \). (c) When (in seconds) will the rock hit the surface? (Round your answer to one decimal place.) ( t . \( \mathrm{m} / \mathrm{s} \) (d) With what velocity (in \( \mathrm{m} / \mathrm{s} \) ) will the rock hit the surface? (d \( \mathrm{m} / \mathrm{s} \)

**(a) Finding the velocity after 2 seconds** The height function is given by \[ H(t)=15t-1.86t^2. \] The velocity is the derivative of the height with respect to time: \[ v(t)=H'(t)=\frac{d}{dt}(15t-1.86t^2)=15-3.72t. \] At \( t=2 \) seconds, \[ v(2)=15-3.72(2)=15-7.44=7.56 \, \text{m/s}. \] **(b) Finding the velocity when \( t=a \)** Using the velocity function, \[ v(t)=15-3.72t, \] when \( t=a \) the velocity is \[ v(a)=15-3.72a. \] **(c) Determining when the rock hits the surface** The rock hits the surface when \( H(t)=0 \): \[ 15t-1.86t^2=0. \] Factor out \( t \): \[ t(15-1.86t)=0. \] This gives two solutions: \[ t=0 \quad \text{or} \quad 15-1.86t=0. \] We discard \( t=0 \) because it represents the initial time. Solving for \( t \) in the second equation: \[ 1.86t=15 \quad \Rightarrow \quad t=\frac{15}{1.86}. \] Calculating the value, \[ t\approx8.0645 \, \text{seconds}. \] Rounded to one decimal place, the rock hits the surface at approximately \[ t\approx8.1 \, \text{seconds}. \] **(d) Finding the velocity when the rock hits the surface** We use the velocity function: \[ v(t)=15-3.72t. \] Substitute \( t=\frac{15}{1.86} \) into the equation: \[ v\left(\frac{15}{1.86}\right)=15-3.72\left(\frac{15}{1.86}\right). \] Notice that \[ \frac{3.72}{1.86}=2, \] so the equation simplifies to \[ v\left(\frac{15}{1.86}\right)=15-2(15)=15-30=-15 \, \text{m/s}. \]