To find the greatest length x such that 3.5m and 8.75m are integral multiples of x, we need to first understand what is meant by integral multiples and greatest common multiple.



Integral multiples refer to a whole number that is multiplied by a given quantity to obtain another whole number. For example, 2 is an integral multiple of 4 because 2 multiplied by 4 gives 8, which is a whole number.



The greatest common multiple is the largest number that is a multiple of two or more given numbers. For example, the greatest common multiple of 6 and 8 is 24 because it is the smallest number that is divisible by both 6 and 8.



To find the greatest length x such that 3.5m and 8.75m are integral multiples of x, we need to first convert the decimals to fractions.

3.5m can be written as 7/2m (since 3.5 is half of 7)
8.75m can be written as 35/4m (since 8.75 is 35/4)

Now, we need to find the least common multiple (LCM) of the two fractions. To do this, we first find the prime factorization of each fraction.

7/2 = 7 * 1/2
35/4 = 5 * 7/2 * 1/2

Next, we take the highest power of each prime factor and multiply them together to get the LCM.

LCM = 5 * 7 * 1/2 * 1/2
LCM = 17.5m

Therefore, the greatest length x that is an integral multiple of both 3.5m and 8.75m is 17.5m.