Nikolai can spend up to 7 hours this weekend performing chores around his parents’ veterinary practice. They pay him $3 for every kennel he cleans and $10 for every dog he bathes. It takes Nikolai 10 minutes to clean out a kennel and 30 minutes to bathe a dog. There are currently 30 dirty kennels and 8 dogs that need to be bathed. Part A: How many of each chore should Nikolai complete in order to earn the most money? Part B: Nikolai dislikes bathing dogs, so he decides to start by cleaning all of the kennels first. He will then bathe dogs with any time that remains. If he begins with the kennels, how many dogs will he have time left to bathe?

We start by setting out the inequalities

Let the number of kennels is 'k' and the number of dogs is 'd'

Inequality 1:

30 minutes per dog + 10 minutes per kennel ≤ 7 hours
30d + 10k ≤ (7 × 60)
30d + 10k ≤ 420

Inequality 2:

Number of kennels to be cleaned is 30
k ≤ 30

Inequality 3:

Number of dogs to bathe is 10
d ≤ 10

Sketching the graph on the Cartesian axes will look like the diagram below
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PART A

From the graph, we can see two intersection coordinates that satisfy the three inequality, which we will check in turn for the greatest possible money that Nikolai could earn

First coordinate (18, 8) ⇒ This means, 18 kernels and 8 dogs
(18 × $3) + (8 × $10) = 54 + 80 = $134

Second coordinate (30, 4) ⇒ This means, 30 kernels and 4 dogs
(30 × $3) + (4 × $10) = $90 + $40 = $130

The maximum earning Nikolai could get is $134 if he cleans 18 kernels and baths 8 dogs.
-----------------------------------------------------------------------------------------------------------PART B

If Nikolai cleans all the kernels first, he'd spend 30 × 10 minutes = 300 minutes out of 420 minutes he has in total in a day

He'd have 420 - 300 = 120 minutes left to bath some dogs.

He needs 30 minutes to bath one dog
With 120 minutes he's got left, he can only bath 120 ÷ 30 = 4 dogs