Question

The deck of a bridge is suspended 275 feet above a river. If a pebble falls of the side of the bridge, the height, in feet of the pebble above the water surface after t seconds is given by\(y = {\bf{275}} - {\bf{16}}{t^{\bf{2}}}\)

(a) Find the average velocity of the pebble for the time period beginning when\(t = {\bf{4}}\)and lasting

(i) 0.1 seconds (ii) 0.05 seconds (iii) 0.01 seconds

(b) Estimate the instaneous velocity of pebble after 4 seconds

Short answer

(a) (i) \( - 129.6\;{\rm{ft/s}}\)

(ii) \( - 128.8\;{\rm{ft/s}}\)

(iii) \( - 128.16\;{\rm{ft/s}}\)

(b) The instantaneous velocity of pebble is \( - 128\;{\rm{ft/s}}\).

Step-by-step solution

Step 1:Find answer for part (a)

Substitute 4 for \(t\) in equation \(y = 275 - 16{t^2}\).

\(\begin{aligned}{c}y &= 275 - 16\left( {{4^2}} \right)\\ &= 19\end{aligned}\)

The expression for average velocity can be calculated as follows:

\(\begin{aligned}{c}{v_{avg}} &= \frac{{y\left( {4 + h} \right) - y\left( 4 \right)}}{{\left( {4 + h} \right) - 4}}\\ &= \frac{{\left( {275 - 16{{\left( {4 + h} \right)}^2}} \right) - 19}}{h}\\ &= \frac{{ - 128h - 16{h^2}}}{h}\\ &= - 128 - 16h\end{aligned}\)

(i) Substitute 0.1 for \(h\) in the equation \({v_{avg}} = - 128 - 16h\).

\(\begin{aligned}{c}{v_{avg}} &= - 128 - 16\left( {0.1} \right)\\ &= - 129.6\;{\rm{ft/s}}\end{aligned}\)

(ii) Substitute 0.05 for \(h\) in the equation \({v_{avg}} = - 128 - 16h\).

\(\begin{aligned}{c}{v_{avg}} &= - 128 - 16\left( {0.05} \right)\\ &= - 128.8\;{\rm{ft/s}}\end{aligned}\)

(iii) Substitute 0.01 for \(h\) in the equation \({v_{avg}} = - 128 - 16h\).

\(\begin{aligned}{c}{v_{avg}} &= - 128 - 16\left( {0.01} \right)\\ &= - 128.16\;{\rm{ft/s}}\end{aligned}\)

Find the answer for part (b)

As t is approaching to 4, the instantaneous velocity of pebble is \( - 128\;{\rm{ft/s}}\).