In the $\boldsymbol{x y}$-plane, a line with equation $\mathbf{2 y}=\mathbf{4 . 5}$ intersects a parabola at exactly one point. If the parabola has equation $\mathbf{y}=-\mathbf{4 x ^ { 2 } }+\mathbf{b x}$, where $\mathbf{b}$ is a positive constant, what is the value of $\mathbf{b}$ ?

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: First, we need to find the point of intersection between the line and the parabola. : Now, we will substitute this value of $y$ into the equation of the parabola, $y = -4x^2 + bx$, to find the corresponding value of $x$: : To solve for $x$, we first rewrite the equation in standard form by bringing all terms to the left side: : Since the parabola intersects the line at exactly one point, this quadratic equation has only one real root. : Expanding and simplifying the equation above, we get: : Taking the square root of both sides, we find the value of $b$: The value of $b$ is 6.

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QuestionAnatomy and Physiology

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Step 1
: First, we need to find the point of intersection between the line and the parabola.

Step 2
: Now, we will substitute this value of $y$ into the equation of the parabola, $y = - 4x^2 + bx$, to find the corresponding value of $x$:

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