Two chords AB and CD intersect internally at X. Given that AX = 10cm, XB = 8cm and CD = 21cm, determine the length XD

When two chords AB and CD intersect internally at a point X, the product of the segments of one chord equals the product of the segments of the other chord.

Given $AX = 10 \text{ cm}$, $XB = 8 \text{ cm}$, and $CD = 21 \text{ cm}$.
Let $XD = x$. Then $CX = CD - XD = 21 - x$.

Rearrange the equation into standard quadratic form $ax^2 + bx + c = 0$:

We can factor the quadratic equation. We need two numbers that multiply to 80 and add up to -21. These numbers are -5 and -16.

This gives two possible values for $x$:

Both values are positive and physically possible lengths.
If $XD = 5 \text{ cm}$, then $CX = 21 - 5 = 16 \text{ cm}$.
If $XD = 16 \text{ cm}$, then $CX = 21 - 16 = 5 \text{ cm}$.
Both solutions satisfy the theorem ($10 \cdot 8 = 80$ and $5 \cdot 16 = 80$).

The length XD can be either $\boxed{\text{5 cm}}$ or $\boxed{\text{16 cm}}$.

15. Calculate the area bounded by the curve $y = x^2 + 2x$, the lines $x = 1, x = 3$ and the x-axis.

The area $A$ bounded by the curve $y = f(x)$, the x-axis, and the lines $x=a$ and $x=b$ is given by $\int_{a}^{b} f(x)\,dx$.
For $y = x^2 + 2x$ in the interval $[1, 3]$, the function is positive, so the area is directly given by the integral.

The area is $\boxed{\frac{50}{3} \text{ square units}}$.

16. Determine the coordinate of point A.

Point P has coordinates $(8, 4, -1)$, so its position vector is $\vec{p} = 8\vec{i} + 4\vec{j} - \vec{k}$.
Point P divides line AB internally in the ratio 4:1, meaning $AP:PB = 4:1$. So, $m=4$ and $n=1$.
The position vector of point B is $\vec{b} = \vec{i} + 3\vec{j} - 2\vec{k}$.
Let the position vector of point A be $\vec{a} = x\vec{i} + y\vec{j} + z\vec{k}$.

The formula for a point P dividing a line segment AB in the ratio $m:n$ is:

Substitute the known values:

Multiply both sides by 5:

Isolate $\vec{a}$:

The position vector $\vec{a} = 36\vec{i} + 8\vec{j} + 3\vec{k}$ corresponds to the coordinate $(36, 8, 3)$.

The coordinate of point A is $\boxed{(36, 8, 3)}$.