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Explanation:
Please Show Work So I Can Grasp The Concept
The tangent line to the graph of y=g(x) at x=4 has equation y=-3x+11. What is the equation of the tangent line to the graph of y=g(x)^3 at x=4? Need correct answer and explanation as soon as possible! Will give brainliest!
Answer:
The equation of the tangent of g(x)^3 at x = 4 is y = 3 - x
Explanation:
The tangent of y = g(x) = -3·x + 11
Therefore, the slope of g(x) = 1/3
The value of y = -3*4 + 11 = -1
The equation of the line g(x) is given as follows;
y - 1 = 1/3*(x - 4)
y - 1 = 1/3x - 4/3
y = 1/3x - 4/3 + 1 = 1/3x - 1/3
g(x) = 1/3x - 1/3
g(x)^3 = (1/3x - 1/3)^3 = [tex]\dfrac{x^3 -3\cdot x^2 + 3 \cdot x - 1}{27}[/tex]
The slope is therefore;
[tex]\dfrac{\mathrm{d} g(x)^{3}}{\mathrm{d} x} = \dfrac{27 \cdot (3 \cdot x^2 -6\cdot x +3 )}{729}[/tex]
The slope of the tangent is the negative reciprocal of the slope of the line which gives;
[tex]Slope \ of \ tangent \ of \ g(x)^3= -\dfrac{729}{27 \cdot (3 \cdot x^2 -6\cdot x +3 )} = -\dfrac{9}{x^2 -2\cdot x + 1}[/tex]
The value of the slope at x = 4 is [tex]-\dfrac{9}{4^2 -2\cdot 4 + 1} = \dfrac{-9}{9} = -1[/tex]
Therefore, we have;
y at x = 4
[tex]y = \dfrac{4^3 -3\cdot 4^2 + 3 \cdot 4 - 1}{27} = \dfrac{27}{27} = 1[/tex]
Therefore, the equation of the tangent is given as follows;
y - 1 =(-1) × (x - 4) = 4 - x
y = 4 - 1 - x = 3 - x
The equation of the tangent of g(x)^3 at x = 4 is y = 3 - x.
Clint wants to keep a young tree from blow- ing over during a high wind. He decides that he should tie a rope around the tree at a height of 180 cm, and then tie the rope to a stake 200 cm from the base. Allowing 30 cm of rope for doing the tying, will the 270-cm rope he has be long enough to do the job?
Answer:
No, the 270 cm rope Clint has is not enough to do the job.
Explanation:
The height of tree where rope has to be tied = 180 cm
The distance of the stake from the base of the tree = 200 cm
We can say that the arrangement makes a right angle triangle, with the right angle situated at the base of the tree. The figure is attached below. Consult it for better understanding.
Length of the rope can be found by using Pythagoras theorem.
(Length of rope)² = 180² + 200²
Length of rope = 269 cm
As it is stated the additional 30cm of rope is required for tying from both ends. Total length of rope required is:
Total length of rope = 269 + 30
Total length of rope = 299 cm
As length of rope required is 299cm, the 270 cm rope Clint has is not enough to do the job.