Mouse A and Mouse B are separated by a distance of 1.62 meters underground. They decide to meet by digging all the way through. Mouse A will double his speed every day, that is, he starts to dig 2cm the first day, 4cm the second day, and so on. Mouse B will dig at a constant speed, that is, 6cm per day. How many centimeters will Mouse A have dug when they finally meet?
I tried to find the number of days it takes by:
$162 = 2^x + 6x$
I couldn't find any integer solutions, but I can't find my error. Help? Thanks.
$2^x$ is the speed of mouse A on day $x$, rather than the distance travelled, which is $$2+4+\cdots+2^x = 2^{x+1}-2.$$ Hence your equation should be $162 = 2^{x+1}-2+6x$, which is solved by $x=6$.
Thus, the distance travelled by mouse A will be $2^{6+1}-2 = 126$.