Submission $1(3\mathrm{.}75/6\mathrm{.}25$ points$)$Monday$,$ July $28,202506$:$40$ PM EDT LarCAApCalc$28\mathrm{.}4\mathrm{.}048.$ A fast$-$food restaurant determines the cost and revenue models for its hamburgers. C$=0\mathrm{.}5$x $+7100,0$ x $50,000$R$=110,000(64,000$x $$ x$2),0$ x $50,000($a$)$ Write the profit function for this situation. P $=$ $$ $($b$)$ Determine the intervals on which the profit function is increasing and decreasing. $($Enter your answer using interval notation.$)$ increasing$$ $$ decreasing$$ $$ $($c$)$ Determine how many hamburgers the restaurant needs to sell to obtain a maximum profit. $29500$ hamburgers Explain your reasoning. Because the function is always decreasing, the maximum profit occurs at this value of x$.$Because the function changes from increasing to decreasing at this value of x$,$ the maximum profit occurs at this value.$$Because the function changes from decreasing to increasing at this value of x$,$ the maximum profit occurs at this value.Because the function is always increasing, the maximum profit occurs at this value of x$.$The restaurant makes the same amount of money no matter how many hamburgers are sold.