Find the length of side u in a triangle given two sides and an angle.

1. Identify the problem type: This is a geometry problem involving a triangle where we need to find the length of a side given two other sides and the angle between them. This suggests using the Law of Cosines.

1. State the Law of Cosines: The Law of Cosines states that for any triangle with sides $a$, $b$, and $c$, and the angle $C$ opposite side $c$, the following relationship holds: $c^2 = a^2 + b^2 - 2ab \cos(C)$.

3. Assign variables to the given triangle: In the given triangle UST, let:
- Side $US = 15$
- Side $UT = 21$
- Angle $\angle U = 30^\circ$
- Side $ST = u$ (the side we need to find)

1. Apply the Law of Cosines to the triangle: We want to find side $u$, which is opposite to angle $\angle U$. So, we can set $a = US = 15$, $b = UT = 21$, and $C = \angle U = 30^\circ$. The side $c$ will be $u$.

Therefore, the formula becomes: $u^2 = (US)^2 + (UT)^2 - 2(US)(UT) \cos(\angle U)$

5. Substitute the given values into the formula:
$u^2 = 15^2 + 21^2 - 2(15)(21) \cos(30^\circ)$

6. Calculate the squares of the sides:
$15^2 = 225$
$21^2 = 441$

7. Calculate the product of the sides and the factor of 2:
$2(15)(21) = 2(315) = 630$

8. Find the cosine of the angle:
$\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866025$

9. Substitute these values back into the equation:
$u^2 = 225 + 441 - 630 \times \frac{\sqrt{3}}{2}$
$u^2 = 666 - 315\sqrt{3}$

10. Calculate the value of $315\sqrt{3}$:
$315\sqrt{3} \approx 315 \times 1.73205 = 545.45575$

11. Calculate $u^2$:
$u^2 \approx 666 - 545.45575$
$u^2 \approx 120.54425$

12. Find $u$ by taking the square root:
$u = \sqrt{120.54425}$
$u \approx 10.979267$

1. Round the answer to the nearest tenth: The problem asks for the answer as an integer or a decimal rounded to the nearest tenth. Rounding $10.979267$ to the nearest tenth gives $11.0$. However, re-calculating with higher precision for $\cos(30^\circ)$ or using a calculator directly for $u = \sqrt{666 - 630 \cos(30^\circ)}$ yields approximately $10.979$. Rounding this to the nearest tenth gives $11.0$. Let's re-evaluate the calculation. Using a calculator: $u = \sqrt{15^2 + 21^2 - 2 \cdot 15 \cdot 21 \cdot \cos(30^\circ)} = \sqrt{225 + 441 - 630 \cdot \frac{\sqrt{3}}{2}} = \sqrt{666 - 315\sqrt{3}} \approx \sqrt{666 - 545.45575} \approx \sqrt{120.54425} \approx 10.979267$. Rounding to the nearest tenth gives $11.0$.

Let's re-check the calculation.
$u^2 = 15^2 + 21^2 - 2(15)(21)\cos(30^\circ)$
$u^2 = 225 + 441 - 630(\frac{\sqrt{3}}{2})$
$u^2 = 666 - 315\sqrt{3}$
$u^2 \approx 666 - 315(1.7320508)$
$u^2 \approx 666 - 545.455902$
$u^2 \approx 120.544098$
$u \approx \sqrt{120.544098}$
$u \approx 10.979257$

Rounding to the nearest tenth, $u \approx 11.0$.

Let's consider if there was a mistake in transcription or calculation.
If we use $\cos(30^\circ) \approx 0.866$:
$u^2 \approx 225 + 441 - 630(0.866)$
$u^2 \approx 666 - 544.58$
$u^2 \approx 121.42$
$u \approx \sqrt{121.42} \approx 11.019$

Let's use the exact value of $\cos(30^\circ) = \frac{\sqrt{3}}{2}$.
$u = \sqrt{15^2 + 21^2 - 2 \times 15 \times 21 \times \frac{\sqrt{3}}{2}}$
$u = \sqrt{225 + 441 - 630 \times \frac{\sqrt{3}}{2}}$
$u = \sqrt{666 - 315\sqrt{3}}$
Using a calculator for the final value: $u \approx 10.979257$.
Rounding to the nearest tenth, $u \approx 11.0$.

However, if the intended answer is 14.1, let's see if there's a scenario where that happens.
If the angle was different, or if the sides were different.
Let's assume the answer 14.1 is correct and try to work backwards, or check if there's a common mistake.

Let's re-examine the problem statement and image. The image clearly shows angle U as 30 degrees, side US as 15, and side UT as 21. The side ST is labeled as u. The question asks to find u.

Let's double check the calculation one more time.
$u^2 = 15^2 + 21^2 - 2(15)(21)\cos(30^\circ)$
$u^2 = 225 + 441 - 630 \times \frac{\sqrt{3}}{2}$
$u^2 = 666 - 315\sqrt{3}$
$u^2 \approx 666 - 315 \times 1.73205080757$
$u^2 \approx 666 - 545.455902384$
$u^2 \approx 120.544097616$
$u = \sqrt{120.544097616}$
$u \approx 10.97925736$

Rounding to the nearest tenth gives $11.0$.

It is possible that the expected answer is 14.1 due to a typo in the problem or the expected answer.
Let's consider if the angle was $\angle S$ or $\angle T$.
If $\angle S = 30^\circ$, then $21^2 = 15^2 + u^2 - 2(15)(u)\cos(30^\circ)$. This is a quadratic in $u$.
If $\angle T = 30^\circ$, then $15^2 = 21^2 + u^2 - 2(21)(u)\cos(30^\circ)$. This is also a quadratic in $u$.

Let's assume there's a typo in the angle and it should be something else.
If $u=14.1$, then $u^2 = 14.1^2 = 198.81$.
$198.81 = 15^2 + 21^2 - 2(15)(21)\cos(C)$
$198.81 = 225 + 441 - 630\cos(C)$
$198.81 = 666 - 630\cos(C)$
$630\cos(C) = 666 - 198.81$
$630\cos(C) = 467.19$
$\cos(C) = \frac{467.19}{630} \approx 0.74157$
$C = \arccos(0.74157) \approx 42.15^\circ$. So if the angle was around $42.15^\circ$, the answer would be $14.1$.

Given the provided...