A stone is thrown vertically into the air at an initial velocity of \( 73 \mathrm{ft} / \mathrm{s} \). On a different planet, the height s (in feet) of the stone above the ground after t seconds is \( \mathrm{s}=73 \mathrm{t}-5 \mathrm{t}^{2} \) and on Earth it is \( \mathrm{s}=73 \mathrm{t}-16 \mathrm{t}^{2} \). How much higher will the stone travel on the other planet than on Earth? When an object that is thrown upwards reaches its highest point (just before it starts to fall back to the ground), its maximum heights of the two objects, use the

We begin with the given height functions: - On the other planet: \[ s=73t-5t^2 \] - On Earth: \[ s=73t-16t^2 \] The maximum height occurs when the derivative of \( s \) with respect to \( t \) is zero. --- **Step 1. Differentiate the height functions** For the other planet, differentiate: \[ \frac{ds}{dt}=73-10t. \] Set the derivative equal to zero: \[ 73-10t=0 \quad\Longrightarrow\quad t=\frac{73}{10}. \] For Earth, differentiate: \[ \frac{ds}{dt}=73-32t. \] Set the derivative equal to zero: \[ 73-32t=0 \quad\Longrightarrow\quad t=\frac{73}{32}. \] --- **Step 2. Calculate the maximum heights** The maximum height is obtained by substituting the time calculated back into the corresponding height function. *Other planet's height:* \[ h_{\text{planet}}=73\left(\frac{73}{10}\right)-5\left(\frac{73}{10}\right)^2. \] Rather than compute step‐by‐step, we can notice that for any quadratic of the form \[ s=vt-at^2, \] the maximum height occurs at \[ t=\frac{v}{2a}, \] and the maximum height is given by \[ h_{\text{max}}=\frac{v^2}{4a}. \] Thus, \[ h_{\text{planet}}=\frac{73^2}{4\cdot 5}=\frac{5329}{20}. \] Numerically, \[ h_{\text{planet}}\approx \frac{5329}{20}\approx 266.45\text{ ft}. \] *Earth's height:* \[ h_{\text{earth}}=\frac{73^2}{4\cdot 16}=\frac{5329}{64}. \] Numerically, \[ h_{\text{earth}}\approx \frac{5329}{64}\approx 83.27\text{ ft}. \] --- **Step 3. Compute the difference** Subtract the two maximum heights: \[ \Delta h = h_{\text{planet}} - h_{\text{earth}} = \frac{5329}{20} - \frac{5329}{64}. \] To combine these, factor out \(5329\): \[ \Delta h = 5329\left(\frac{1}{20}-\frac{1}{64}\right). \] Find a common denominator: \[ \frac{1}{20}-\frac{1}{64}=\frac{64-20}{20\cdot 64}=\frac{44}{1280}. \] Thus, \[ \Delta h=5329\cdot\frac{44}{1280}=\frac{5329\cdot44}{1280}. \] Calculating the numerator: \[ 5329\cdot44=234476. \] Thus, \[ \Delta h=\frac{234476}{1280}\approx183.22\text{ ft}. \] --- The stone travels approximately \(183.22\) feet higher on the other planet than on Earth.