Question

Amir throws a stone off of a bridge into a river. The stone's height (in meters above the water) ttt seconds after Amir throws it is modeled by h(t)=-5t^2+20t+160h(t)=−5t 2 +20t+160h, left parenthesis, t, right parenthesis, equals, minus, 5, t, squared, plus, 20, t, plus, 160 Amir wants to know when the stone will reach its highest point. 1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation. h(t)=h(t)=h, left parenthesis, t, right parenthesis, equals 2) How many seconds after being thrown did the stone reach its highest point?

Answer 1

Answer:

1) The vertex form of  $h(t) = -5\cdot t^{2}+20\cdot t +160$ is $h -180 = -5\cdot (t-2)^{2}$, 2) The stone reaches its maximum height 2 seconds after being thrown.

Step-by-step explanation:

1) Given that height of the stone is represented by a second-order polynomial, which depicts a parabola as graph. The best approach to determine the instant when stone reaches its highest is by vertex form, whose form is:

$h-k = C\cdot (t-r)^{2}$

Where:

$r$, $k$ - Instant and maximum height of the stone, measured in seconds and meters.

$C$ - Vertex constant, which must be negative as there is an absolute maximum, measured in meters per square second.

Let be $h(t) = -5\cdot t^{2}+20\cdot t +160$, which is transformed into vertex form:

i) $h = -5\cdot t^{2}+20\cdot t +160$ Given

ii) $h = -5\cdot (t^{2}-4\cdot t -32)$ Distributive property/$(-a)\cdot b = -a\cdot b$

iii) $h = -5\cdot [(t^{2}-4\cdot t +4)+(-36)]$ Existence of additive inverse/Definitions of addition and subtraction

iv) $h = (-5)\cdot (t-2)^{2}+180$ Distributive property/$(-a)\cdot (-b) = a\cdot b$/Perfect square binomial

v) $h -180 = -5\cdot (t-2)^{2}$ Compatibility with addition/Existence of additive inverse/Modulative property/Definition of subtraction/Result

The vertex form of  $h(t) = -5\cdot t^{2}+20\cdot t +160$ is $h -180 = -5\cdot (t-2)^{2}$.

2) The time can be extracted from previous results, which indicates that stone reaches its maximum height 2 seconds after being thrown.