Functional Modeling Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
A ball is dropped from a 100100 m building. Using h(t)=100−4.9t2h(t)=100-4.9t^2, find: (a) height at t=3t=3 s, (b) time to hit the ground, (c) interpret h′(t)h'(t) at impact.

Solution

  1. 1
    (a) h(3)=100−4.9(9)=100−44.1=55.9h(3)=100-4.9(9)=100-44.1=55.9 m.
  2. 2
    (b) Set h(t)=0h(t)=0: 100−4.9t2=0⇒t2=1004.9≈20.41⇒t≈4.52100-4.9t^2=0 \Rightarrow t^2=\frac{100}{4.9}\approx20.41 \Rightarrow t\approx4.52 s.
  3. 3
    (c) h′(t)=−9.8th'(t)=-9.8t. At impact t≈4.52t\approx4.52: h′(4.52)≈−9.8(4.52)≈−44.3h'(4.52)\approx-9.8(4.52)\approx-44.3 m/s. The ball hits at ≈44\approx44 m/s downward.

Answer

(a) 55.955.9 m; (b) t≈4.52t\approx4.52 s; (c) impact velocity ≈−44.3\approx-44.3 m/s
Physics problems become function modeling problems. The quadratic h(t)h(t) captures free fall under gravity. The derivative (velocity) at impact shows how fast the ball is moving — this is instantaneous rate of change applied to motion.

About Functional Modeling

Functional modeling uses functions to represent relationships between real-world quantities — choosing the right function family to capture the observed pattern.

Learn more about Functional Modeling →

More Functional Modeling Examples

Example 1 easy

A rectangular garden has perimeter [formula] m. Express the area [formula] as a function of the widt

Example 3 easy

A taxi charges [formula]2.50[formula][formula] per [formula] mile. Write a cost function [formula] i

Example 4 hard

A cylindrical can must hold [formula] cm³. Express the total surface area [formula] as a function of

← All Functional Modeling Examples Functional Modeling Concept