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bogdanovich [222]
3 years ago
11

The curve with equation y=ax^2+bx has a gradient of 3 at the point (2,-2). Find the values of a and b.

Mathematics
1 answer:
Airida [17]3 years ago
7 0

Answer:

a = 2, b = - 5

Step-by-step explanation:

Given

y = ax² + bx

\frac{dy}{dx} is the measure of the slope at x = a

Differentiate each term with respect to x using the power rule

\frac{d}{dx}(ax^{n} ) = nax^{n-1}

\frac{dy}{dx} = 2ax + b, hence

2ax + b = 3 at (2, - 2)

Substitute x = 2 into \frac{dy}{dx}

4a + b = 3 → (1) and substitute x = 2 into y

4a + 2b = - 2 → (2)

Subtract ( 1) from (2)

b = - 2 - 3 = - 5

Substitute b = - 5 into (1)

4a - 5 = 3 ( add 5 to both sides )

4a = 8 ( divide both sides by 4 )

a = 2

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jolli1 [7]

Answer:

The length of the rectangle (l) = 3 feet

The width of the rectangle (W) =  10 feet

Step-by-step explanation:

<u><em>Step(i):-</em></u>

<em>Given that the width of the rectangle  = (x+1) feet</em>

<em>Given that the length of the rectangle = ( x-6) feet</em>

<em>The area of the rectangle = 30 square feet</em>

<u><em>Step(ii):-</em></u>

We know that the area of the rectangle

                = length ×width

    30   = ( x+1)(x-6)

    30   = x² - 6x + x -6

⇒ x² - 5 x - 6 = 30

⇒ x² - 5 x - 6 - 30 =0

⇒ x² - 5 x - 36 =0

 x² - 9 x +4x - 36 =0

   x (x-9) +4 ( x-9) =0

   ( x+4 ) ( x-9) =0

     ( x+4 ) =0 and  ( x-9) =0

     x =-4 and x =9

<u><em>Step(iii):-</em></u>

we have to choose x =9

     The length of the rectangle (l) =  x-6 = 9-6 =3

      The width of the rectangle (W) =  x+1 =  9 +1 = 10

<u><em>Final answer:-</em></u>

The length of the rectangle (l) = 3 feet

The width of the rectangle (W) =  10 feet

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