If the maximum value of x+y on the set {(x,y)∣x≥0,y≥0,x+5y≤10,5x+y≤a2−5} is 5, then the value of a=…
Explanation
Given the system of inequalities:
⎩⎨⎧x≥0y≥0x+5y≤105x+y≤a2−5
We can sum the two main inequalities to find the bound for x+y:
(x+5y)+(5x+y)≤10+(a2−5)
6x+6y≤5+a2
6(x+y)≤5+a2
It is given that the maximum value of x+y is 5. Therefore, we substitute this maximum value into the inequality:
6(5)=5+a2
30=5+a2
a2=25
Thus, the value of a is:
a=5
(We take the positive value since the options are positive).
So, the value of a is 5.