Answer:
y=6.15cm
total length covered 9.12cm
Step-by-step explanation:
From the equation of a parabola
Recall that:
Parabola->
y=ax^2+bx+c
& y= height rabbit jumps & x= length rabbit jumps
What is max height rabbit can jump?
Solving for the x coordinate of the vertex:
x=−b/2a
from y=-0.296x^2+2.7x
Let a= -0.296 & b=2.7
x=−2.7/2(−0.296)
x=4.56
To find y coordinate when x= 4.56
... y=-0.296x^2+2.7x
-0.296(4.56)^2+2.7(4.56)
y=6.15cm
The total length covered will be 9.12cm=2x
The maximum length and the height of the jump by my sister's rabbit are [tex]9.12\;\rm cm[/tex] and [tex]4.56\;\rm cm[/tex] respectively.
The given function is [tex]y=-0.296x^2+2.7x[/tex] where, [tex]x[/tex] is the length of the jump and [tex]y[/tex] is the height of the jump by my sister's rabbit in centimeters.
Differentiate the given equation with respect to [tex]x[/tex] to evaluate the maximum length of the jump as-
[tex]\dfrac{dy}{dx}=\dfrac{d}{dx}(-0.296x^2+2.7x)\\\dfrac{dy}{dx}=(-0.296\times 2)x+2.7[/tex]
Differentiate the above equation with respect to [tex]x[/tex] again,
[tex]\dfrac{d^2y}{dx^2}=-0.296\times 2[/tex] which is negative.
So, the value of [tex]x[/tex] will give the maximum value.
Equate [tex]\dfrac{dy}{dx}=0[/tex] to get the maximum value of [tex]x[/tex] as-
[tex]\dfrac{dy}{dx}=0\\(-0.296\times 2)x+2.7=0\\x=4.56\;\rm cm[/tex]
For the total length of the rabbit's jump,
[tex]y=2\times 4.56\\y=9.12\;\rm cm[/tex]
Hence, the maximum length and the height of the jump by my sister's rabbit are [tex]9.12\;\rm cm[/tex] and [tex]4.56\;\rm cm[/tex] respectively.
Learn more about differentaition here:
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