What will be the estimate error made in calculating the area of the triangle ABC in which the sides a and b are measured accurately as 25 cm and 16 cm, while the angle C is measured as 60° but (1/2)° in error?

The correct option is 55/63 sq cm.

To find the estimated error in calculating the area of the triangle, we start with the formula for the area of a triangle when two sides and their included angle are known. The area of the triangle $ABC$, denoted as $\Delta$, is given by:

$$\Delta =\frac{1}{2}ab\sin C$$

We are given that the sides $a$ and $b$ are measured accurately. This means their errors are zero:
$da=0$ and $db=0$.
Therefore, the error in the calculated area, $d\Delta$, depends solely on the error in the measurement of the angle $C$, denoted as $dC$.

Using calculus, the differential error in the area is obtained by differentiating $\Delta$ with respect to $C$:

$$d\Delta =\frac{\partial \Delta }{\partial C}dC=\frac{1}{2}ab\cos CdC$$

Now, let's list the given values from the problem:
• $a=25\text{ cm}$
• $b=16\text{ cm}$
• $C=60°$
• The error in angle measurement, $dC={\left(\frac{1}{2}\right)}^{°}$

Before substituting $dC$ into our formula, we must convert it from degrees to radians, because trigonometric derivatives assume angles are in radians:

$$dC=\frac{1}{2}\times \frac{\pi }{180}=\frac{\pi }{360}\text{ radians}$$

Now we substitute all the values into the error formula:

$$d\Delta =\frac{1}{2}\times 25\times 16\times \cos (60°)\times \frac{\pi }{360}$$

Since we know that $\cos (60°)=\frac{1}{2}$, the equation simplifies to:

$$d\Delta =\frac{1}{2}\times 400\times \frac{1}{2}\times \frac{\pi }{360}$$

$$d\Delta =100\times \frac{\pi }{360}=\frac{5\pi }{18}$$

Using the standard rational approximation for $\pi \approx \frac{22}{7}$:

$$d\Delta \approx \frac{5}{18}\times \frac{22}{7}=\frac{110}{126}=\frac{55}{63}\text{ sq cm}$$

Thus, the estimated error made in calculating the area of the triangle is 55/63 sq cm.