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Ricardo throws a stone off a bridge into a river below. The stone's height (in meters above the water), xxx seconds after Ricardo threw it, is modeled by w(x)=-5(x-8)(x+4)w(x)=−5(x−8)(x+4)w, left parenthesis, x, right parenthesis, equals, minus, 5, left parenthesis, x, minus, 8, right parenthesis, left parenthesis, x, plus, 4, right parenthesis What is the maximum height that the stone will reach?

Respuesta :

carlosego
For this case we have the following function:
 w (x) = - 5 (x-8) (x + 4)
 Rewriting we have:
 w (x) = - 5 (x ^ 2 + 4x - 8x - 32)
 w (x) = - 5x ^ 2 - 20x + 40x + 160
 w (x) = - 5x ^ 2 + 20x + 160
 Then, deriving we have:
 w '(x) = - 10x + 20
 We equal zero and clear x:
 0 = -10x + 20
 10x = 20
 x = 20/10
 x = 2 seconds
 Substituting values:
 w (2) = - 5 (2-8) (2 + 4)
 w (2) = - 5 (-6) (6)
 w (2) = 180 meters
 Answer:
 The maximum height that the stone will reach is:
 w (2) = 180 meters

The maximum height at which the stone will reach, when Ricardo throws it off a bridge into a river below, is 180 meters.

What is the equation of parabola?

An equation of parabola is the equation in which the unknown variable is one and the highest power of the unknown variable is two.

The standard form of the equation of parabola is,

ax^2+bx+c=0

Here,(a,b, c) is the real numbers and (x) is the variable.

Ricardo throws a stone off a bridge into a river below.  The stone's height (in meters above the water), x seconds after Ricardo threw it, is modeled by,

[tex]w(x)=-5(x-8)(x+4)[/tex]

Simplify the equation,

[tex]w(x)=-5(x-8)(x+4)\\w(x)=(-5x+40)(x+4)\\w(x)=-5x^2-20x+40x+160\\w(x)=-5x^2+20x+160[/tex]

This is the equation of parabola. Compare it with standard equation, we get,

[tex]a=-5\\b=20\\c=160[/tex]

Thus, the vertex point of this parabola is,

[tex]\dfrac{-b}{2a}=\dfrac{-20}{2\times(-5)}\\\dfrac{-b}{2a}=2[/tex]

At this point, we get the maximum height,

[tex]w(2)=-5(2)^2+20(2)+160\\w(2)=-20+40+160\\w(2)=180\rm\; m[/tex]

Thus, the maximum height at which the stone will reach, when Ricardo throws it off a bridge into a river below, is 180 meters.

Learn more about the quadratic equation here;

https://brainly.com/question/1214333

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