Variance of a random variable measures how spread out the values of the random variable are around the expected value. In this case, we are given that the random variable is uniformly distributed over the interval 2 to 10.

The uniform distribution is characterized by a constant probability density function over a given interval. In this case, the interval is from 2 to 10. The probability density function for a uniform distribution is given by:

f(x) = 1/(b-a), for a≤x≤b

where a is the lower limit of the interval (2 in this case) and b is the upper limit of the interval (10 in this case).

The expected value of a uniform distribution is given by the formula:

E(X) = (a+b)/2

In this case, the expected value of the random variable is (2+10)/2 = 6.

To calculate the variance, we need to find the squared deviation of each value of the random variable from its expected value, and then take the average of these squared deviations.

Variance formula:

Var(X) = E((X-E(X))^2)

Substituting the values, we have:

Var(X) = E((X-6)^2)

Since the random variable is uniformly distributed over the interval 2 to 10, the squared deviation will be the same for all values in this interval.

Calculating the squared deviation for a value x in the interval:

(X-6)^2 = (x-6)^2

To find the expected value, we need to integrate this squared deviation over the interval 2 to 10 and divide by the length of the interval (10-2 = 8):

E((X-6)^2) = (1/8)∫[(x-6)^2]dx from 2 to 10

Simplifying the integral:

E((X-6)^2) = (1/8)∫[(x^2 - 12x + 36)]dx from 2 to 10

= (1/8)[(x^3/3 - 6x^2 + 36x)] from 2 to 10

= (1/8)[(100/3 - 600 + 360) - (8/3 - 24 + 72)]

= (1/8)[(100/3 - 600 + 360) - (8/3 - 72)]

= (1/8)[(-200/3 + 360) - (-56/3)]

= (1/8)[(-200/3 + 360) + (56/3)]

= (1/8)[(-200 + 1080) + 56]

= (1/8)[880 + 56]

= (1/8)(936)

= 117/4

Simplifying:

Var(X) = 117/4

= 29.25

Therefore, the correct answer is option A, 16/3.