Question

When the measured number 0.0090 is multiplied by the measured number 87.10, the answer has two significant figures?

Answer 1

Correct.
The number 0.0090 has two significant figures in it, as we can represent the placeholder zeros (the ones between the decimal and the first 'significant' number - in this case, the 9) with a power of ten: 9.0x10^-3. Additionally, it is a measured number, meaning whatever instrument was used to measure displayed the zero in the ten thousandths place. 
The number 87.10 has four significant figures. There are no placeholder zeros between the decimal and a 'significant' number, and it is also a measurement made by an instrument which displayed a zero in the hundredths place.
When numbers are multiplied or divided, the answer should be rounded to the same number of significant figures as the value with the least number of sig figs in the calculation.
0.0090 x 87.10 = 0.7839
= 0.78  rounded to 2 sig figs
or
7.8 x 10^-1 in scientific notation