The slope of the tangent to the curve y^3x + y^2x^2 = 6 at (2,1)

1 answer:

7 0

Y^3x + y^2x^2 = 6

Using implicit differentiation 

3 y^2.y'.x+y^3+2 y.y'.x^2+2 y^2.x=0

y'(3 y^2.x+2 y. x^2) = -y^3-2 y^2.x 

y' = -[y^2(y+2 x)]/[x y(3 y+2 x)] 

= -[y(y+2 x)]/[x(3 y+2 x)] 

Substituting x =2 and y = 1:

y' = -(1)(1+4)/[2(3+4)] 
 = -5/14 

The slope of the tangent of the curve at (2,1) will be:
m = -5/14

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The pitch of the roof is given by the ratio of the rise to the run of the roof, which is the slope of the rafter

$The \ pitch \ of \ the \ roof \ is \ \mathbf {\dfrac{6}{13}}$

Please find attached the simple diagram of the house with all measurements labeled

The reason the above value for the pitch and the diagram are correct are as follows:

The given parameters are;

Width of the section of the house on which the gable of the roof is to be placed = 50 feet

Length of the over hang on the sides of the roof = 1 foot each

Total span of the roof = 52 feet

Horizontal length of each rafter, the run = 26 feet

The height of the ridge above the top of the frame, the rise = 12 feet

Required:

To find the pitch of the roof

To draw a diagram of the house showing measurements

Solution:

$Pitch = \dfrac{Rise}{Run}$

$Pitch \ (slope) \ of \ roof = \dfrac{12}{26} =\dfrac{6}{13}$

Please find attached the diagram of the house showing the measurements for the roof parameters and the section of the house where the gable roof is to be placed

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