Differentiation

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Question 31

A robotic welding arm follows a path $C\,$ in a horizontal workspace, defined by the parametric equations

\frac{1}{2}t^2 + 2, \quad y = 2t - \frac{8}{t}, \quad t > 0

where $x\,$ and $y\,$ are coordinates in centimetres. The path $C\,$ intersects the $x$ -axis at the point $Q$ .

Determine the $x$ -coordinate of $Q$ .

[2]

A safety barrier is represented by the line $l$ , which is the normal to the path $C\,$ at the point $P$ . Given that $t = 4$ at $P$ :

Write down the coordinates of $P$ .

[1]

Using calculus, show that an equation of $l\,$ is

8 x + 5 y = 110

[4]

The region $R\,$ is bounded by the path $C\,$ from $Q\,$ to $P$ , the line $l\,$ from $P\,$ to the $x$ -axis, and the $x$ -axis between $Q\,$ and the line intercept.

Using algebraic integration, find the exact volume of the solid of revolution formed when the region $R\,$ is rotated through $2\pi\,$ radians about the $x$ -axis.

[7]

Differentiation

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Question 31

A robotic welding arm follows a path $C\,$ in a horizontal workspace, defined by the parametric equations

\frac{1}{2}t^2 + 2, \quad y = 2t - \frac{8}{t}, \quad t > 0

where $x\,$ and $y\,$ are coordinates in centimetres. The path $C\,$ intersects the $x$ -axis at the point $Q$ .

Determine the $x$ -coordinate of $Q$ .

[2]

A safety barrier is represented by the line $l$ , which is the normal to the path $C\,$ at the point $P$ . Given that $t = 4$ at $P$ :

Write down the coordinates of $P$ .

[1]

Using calculus, show that an equation of $l\,$ is

8 x + 5 y = 110

[4]

The region $R\,$ is bounded by the path $C\,$ from $Q\,$ to $P$ , the line $l\,$ from $P\,$ to the $x$ -axis, and the $x$ -axis between $Q\,$ and the line intercept.

Using algebraic integration, find the exact volume of the solid of revolution formed when the region $R\,$ is rotated through $2\pi\,$ radians about the $x$ -axis.

[7]

Differentiation

Question 31

Markscheme

Differentiation

Question 31

Markscheme

Differentiation

Question 31

Differentiation Questions

Differentiation

Question 31

Differentiation Questions