The cross-section of a parabolic archway is modeled by the curve $C\,$ with equation $y = 6x - x^2$ , where $y\,$ is the vertical height in metres and $x\,$ is the horizontal distance from the left-hand base $O\,$ at the origin. A straight support strut $L\,$ is to be installed from the origin to a point $Q\,$ on the archway. The strut follows the equation $y = k x$ , where $k\,$ is a constant and $0 < k < 6$ .

Find, in terms of $k$ , the coordinates of $Q$ .

The region $R_1\,$ is the area of the archway cross-section that lies above the support strut and below the curve $C$ .

Show that the area of $R_1\,$ is

\frac{(6 - k)^3}{6}

The region $R_2\,$ is the area of the archway cross-section that lies below the support strut, above the horizontal ground ( $x$ -axis), and is bounded to the right by the archway.

Given that the area of $R_1\,$ is equal to the area of $R_2$ , find the exact value of $k$ .

Find, in terms of $k$ , the coordinates of $Q$ .

The region $R_1\,$ is the area of the archway cross-section that lies above the support strut and below the curve $C$ .

Show that the area of $R_1\,$ is

\frac{(6 - k)^3}{6}

The region $R_2\,$ is the area of the archway cross-section that lies below the support strut, above the horizontal ground ( $x$ -axis), and is bounded to the right by the archway.

Given that the area of $R_1\,$ is equal to the area of $R_2$ , find the exact value of $k$ .

1.8 Integration

Question 93

Markscheme

1.8 Integration

Question 93

Markscheme

1.8 Integration

Question 93

1.8 Integration Questions

1.8 Integration

Question 93

1.8 Integration Questions