0000:00:00:00:00Start5Number 5ExplanationIf the maximum value of x+yx + yx+y on the set {(x,y)∣x≥0,y≥0,x+5y≤10,5x+y≤a2−5}\{(x,y)|x \ge 0, y \ge 0, x + 5y \le 10, 5x + y \le a^2 - 5\}{(x,y)∣x≥0,y≥0,x+5y≤10,5x+y≤a2−5} is 555, then the value of a=…a = \dotsa=…333444555161616252525ExplanationGiven the system of inequalities: {x≥0y≥0x+5y≤105x+y≤a2−5\begin{cases} x \ge 0 \\ y \ge 0 \\ x + 5y \le 10 \\ 5x + y \le a^2 - 5 \end{cases}⎩⎨⎧x≥0y≥0x+5y≤105x+y≤a2−5 We can sum the two main inequalities to find the bound for x+yx + yx+y: (x+5y)+(5x+y)≤10+(a2−5)(x + 5y) + (5x + y) \le 10 + (a^2 - 5)(x+5y)+(5x+y)≤10+(a2−5)6x+6y≤5+a26x + 6y \le 5 + a^26x+6y≤5+a26(x+y)≤5+a26(x + y) \le 5 + a^26(x+y)≤5+a2 It is given that the maximum value of x+yx + yx+y is 555. Therefore, we substitute this maximum value into the inequality: 6(5)=5+a26(5) = 5 + a^26(5)=5+a230=5+a230 = 5 + a^230=5+a2a2=25a^2 = 25a2=25 Thus, the value of aaa is: a=5a = 5a=5 (We take the positive value since the options are positive). So, the value of aaa is 555.
5Number 5ExplanationIf the maximum value of x+yx + yx+y on the set {(x,y)∣x≥0,y≥0,x+5y≤10,5x+y≤a2−5}\{(x,y)|x \ge 0, y \ge 0, x + 5y \le 10, 5x + y \le a^2 - 5\}{(x,y)∣x≥0,y≥0,x+5y≤10,5x+y≤a2−5} is 555, then the value of a=…a = \dotsa=…333444555161616252525ExplanationGiven the system of inequalities: {x≥0y≥0x+5y≤105x+y≤a2−5\begin{cases} x \ge 0 \\ y \ge 0 \\ x + 5y \le 10 \\ 5x + y \le a^2 - 5 \end{cases}⎩⎨⎧x≥0y≥0x+5y≤105x+y≤a2−5 We can sum the two main inequalities to find the bound for x+yx + yx+y: (x+5y)+(5x+y)≤10+(a2−5)(x + 5y) + (5x + y) \le 10 + (a^2 - 5)(x+5y)+(5x+y)≤10+(a2−5)6x+6y≤5+a26x + 6y \le 5 + a^26x+6y≤5+a26(x+y)≤5+a26(x + y) \le 5 + a^26(x+y)≤5+a2 It is given that the maximum value of x+yx + yx+y is 555. Therefore, we substitute this maximum value into the inequality: 6(5)=5+a26(5) = 5 + a^26(5)=5+a230=5+a230 = 5 + a^230=5+a2a2=25a^2 = 25a2=25 Thus, the value of aaa is: a=5a = 5a=5 (We take the positive value since the options are positive). So, the value of aaa is 555.
