Graphs of two-variable inequalities word problem

dianna the dog receives five dog biscuits for fetching each frisbee and three dog biscuits for fetching each ball it sounds like a pretty good deal she plans to receive at most D dog biscuits before chasing her tail that sounds reasonable the inequality graphed below represents the set of all combinations where Diana fetches F frisbees and fetches B balls in order to receive at most at most D dog biscuits cuz at that point she reasonably starts chasing her tail according to the graph and we're gonna take a look at the graph in a second according to the graph what is the most number of dog biscuits Diana wants to receive before chasing her tail in other words what is d so let's look at let's interpret this graph properly so if we look at the horizontal axis right over here that's F that's the number of frisbees number of frisbees she catches for his B's and this vertical axis this is the number of balls number of number of balls that she gets and we know what the total number of biscuits is are going to be the total from catching frisbees if she catches F frisbees she gets 5 biscuits per frisbee so the total from catching frisbees is 5 F and if she catches if she catches B balls or retrieves B balls so if she gets those B balls and she gets low at three three biscuits per ball yep three dog biscuits for fetching each ball the total number of dog biscuits she gets from fetching B balls is three B and so the total number of biscuits she fetches is five F + 3 B this is the number of biscuits from frisbees this is the number of biscuits from balls and we could see all of the allowable combinations of number of frisbees and number of balls here and so for example if she catches that point right over there eight or retrieves eight balls and catches well that would be half of a frisbee so that doesn't that doesn't seem to make sense but if she if she retrieves eight balls and and catches one frisbee well then then that's still okay she still hasn't met her her maximum number of biscuits yet so how do we think about the nap maximum of biscuits well the mask maximum number of biscuits are any of these points when she that are sitting on this line and notice the solutions that or all of the points that satisfy this inequality are all below this line so she's hitting a maximum when she's on the line and an easy one might be this point right over here where we see that where we see that frisbees zero frisbees and ten and ten balls pretty much maximizes her number of biscuits so if she catches ten balls so let me write this down so if B is equal to 10 B is equal to 10 let me write this F is 0 if F is equal to 0 and B is equal to 10 well how many is she going to catch how or how many biscuits is she going to get well she's going to get this is going to be 0 and then 3 times 10 is 30 this is going to be 30 biscuits 30 biscuits so this point right over here this corresponds with 30 biscuits 30 biscuits this gets and you can see that any of these points along this blue line actually correspond to 30 biscuits if you go over here where f is 6 let me write it here if f is equal to 6 and B is equal to 0 so spends all of her time she earns her biscuits purely through frisbee catching so this is situation where f is equals 6 and B is equal to 0 you still have the same scenario you still have if f is 6 5 times 6 is 30 plus 3 times 0 well that's just going to be once again 30 biscuits 30 biscuits so her maximum or the number of biscuits she needs before she starts chasing her tail is 30 so D is going to be 30 in fact you can express this inequality as saying 5 F plus 3b has to be less than or equal to 30 all right and then they ask us another question can Diana fulfill her plan by fetching can condign up fulfil her plan by fetching 4 frisbees and 2 balls so let's see 4 frisbees and 2 balls this is right over here 4 frisbees and two balls so it's not we're not saying that she has to maximize that she has to get to 30 biscuits she just cannot not eat any more than 30 biscuits so it seems it seems like she can fulfil her plan let me see the inequality graphed below represents all the combinations were diana fetches f frisbees and b balls in order to receive at most D dog biscuits so let's sees is is she fulfilling her plan by fetching well yeah I would say yes