Question

a ball is thrown into the air by a baby alien on a planet in the system of alpha centauri with a velocity of 35 ft/s. its height in feet after t seconds is given by ( y = 35t - 20t^2 ). a. find the average velocity for the time period beginning when t=2 and lasting.01 s:
.005 s:
.002 s:
.001 s:
note: for the above answers, you may have to enter 6 or 7 significant digits if you are using a calculator. estimate the instantaneous velocity when t=2.
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Explanation:

Part A: Average Velocity Calculations

The formula for average velocity over the time interval \([t, t + h]\) is \(\frac{y(t + h) - y(t)}{h}\), where \(y(t) = 35t - 20t^2\) and \(t = 2\).

For \(h = 0.01\) s:

Step 1: Calculate \(y(2 + 0.01)\)

\(t + h = 2.01\)
\(y(2.01) = 35(2.01) - 20(2.01)^2\)
\(= 70.35 - 20(4.0401)\)
\(= 70.35 - 80.802\)
\(= -10.452\)

Step 2: Calculate \(y(2)\)

\(y(2) = 35(2) - 20(2)^2\)
\(= 70 - 80\)
\(= -10\)

Step 3: Calculate average velocity

\(\frac{y(2.01) - y(2)}{0.01} = \frac{-10.452 - (-10)}{0.01}\)
\(= \frac{-0.452}{0.01}\)
\(= -45.2\)

For \(h = 0.005\) s:

Step 1: Calculate \(y(2 + 0.005)\)

\(t + h = 2.005\)
\(y(2.005) = 35(2.005) - 20(2.005)^2\)
\(= 70.175 - 20(4.020025)\)
\(= 70.175 - 80.4005\)
\(= -10.2255\)

Step 2: Calculate average velocity

\(\frac{y(2.005) - y(2)}{0.005} = \frac{-10.2255 - (-10)}{0.005}\)
\(= \frac{-0.2255}{0.005}\)
\(= -45.1\)

For \(h = 0.002\) s:

Step 1: Calculate \(y(2 + 0.002)\)

\(t + h = 2.002\)
\(y(2.002) = 35(2.002) - 20(2.002)^2\)
\(= 70.07 - 20(4.008004)\)
\(= 70.07 - 80.16008\)
\(= -10.09008\)

Step 2: Calculate average velocity

\(\frac{y(2.002) - y(2)}{0.002} = \frac{-10.09008 - (-10)}{0.002}\)
\(= \frac{-0.09008}{0.002}\)
\(= -45.04\)

For \(h = 0.001\) s:

Step 1: Calculate \(y(2 + 0.001)\)

\(t + h = 2.001\)
\(y(2.001) = 35(2.001) - 20(2.001)^2\)
\(= 70.035 - 20(4.004001)\)
\(= 70.035 - 80.08002\)
\(= -10.04502\)

Step 2: Calculate average velocity

\(\frac{y(2.001) - y(2)}{0.001} = \frac{-10.04502 - (-10)}{0.001}\)
\(= \frac{-0.04502}{0.001}\)
\(= -45.02\)

Part B: Instantaneous Velocity Estimate

As \(h\) approaches \(0\), the average velocity approaches the instantaneous velocity. From the calculations above, as \(h\) gets smaller (0.01, 0.005, 0.002, 0.001), the average velocity approaches \(-45\). We can also find the instantaneous velocity by taking the derivative of \(y(t)\) with respect to \(t\).

The derivative of \(y(t) = 35t - 20t^2\) is \(y'(t) = 35 - 40t\).

At \(t = 2\):
\(y'(2) = 35 - 40(2) = 35 - 80 = -45\)

Answer:

s:

• For \(h = 0.01\) s: \(-45.2\)
• For \(h = 0.005\) s: \(-45.1\)
• For \(h = 0.002\) s: \(-45.04\)
• For \(h = 0.001\) s: \(-45.02\)
• Instantaneous velocity at \(t = 2\): \(-45\)