$2223-7215$

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$2.[$Maximum mark: $15]$

Daina makes pendulums to sell at a market. She plans to make $10$ pendulums on the first day and, on each subsequent day, make $6$ more than she did the day before.

$($a$)$ Calculate the number of pendulums she would make on the $12$ th day.

$[3]$

She plans to make pendulums for a total of $15$ days in preparation for going to the market.

$($b$)$ Calculate the total number of pendulums she would have available at the market.

$2$

Daina would like to have at least $1000$ pendulums available to sell at the market and therefore decides to increase her production. She still plans to make $10$ pendulums on the first day, but on each subsequent day, she will make $x$ more than she did the day before.

$($c$)$ Given that she will still make pendulums for a total of $15$ days, calculate the minimum

$3$

integer value of $x$ required for her to reach her target.

Daina tests one of her pendulums. She releases the ball at the end of the pendulum to swing freely. The point at which she releases it is shown as the initial position on the left side of the following diagram. Daina begins recording the distances travelled by the ball after it has reached the extreme position, represented by the right$-$hand side of the diagram.

$($This quest