Math Problem Statement
Question content area top Part 1 If an object is projected upward from ground level with an initial velocity of 160 ft per sec, then its height in feet after t seconds is given by s(t)equalsminus16tsquaredplus160t. Find the number of seconds it will take to reach its maximum height. What is this maximum height?
Solution
To determine the time and maximum height of an object projected upward, we can analyze the height function , where:
- is the time in seconds,
- is the height in feet at time ,
- represents the effect of gravity in feet per second squared,
- represents the initial velocity of 160 feet per second.
Step 1: Determine the time to reach the maximum height
The maximum height of a projectile occurs at the vertex of the parabolic function . For a function of the form , the time at the vertex (or maximum point for a downward-opening parabola) is given by: In our case:
Thus,
So, it takes 5 seconds to reach the maximum height.
Step 2: Calculate the maximum height
Now, substitute into the height function to find the maximum height: [ s(5) = -16(5)^2 +
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Vertex of a Parabola
Projectile Motion
Formulas
Vertex formula for a quadratic function t = -b/(2a)
Height function s(t) = -16t^2 + 160t
Theorems
Vertex Theorem for Quadratic Functions
Suitable Grade Level
Grades 9-10
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