AndInformation we have:
Iron in a banana: 0.3 mg
Iron in a whole vitamin: 4mg
Cole needs at least 11 mg of iron and at most 22 mg.
Part A.
We are told that he eats 3 bananas each day, since each banana has 0.3 mg of Iron, the amount of Iron he gets from bananas is:
[tex]3\times0.3=0.9[/tex]If we call "x" the number of vitamins, we get the following inequality to stay in the limits of Iron per day:
[tex]11\leq4x+0.9\leq22[/tex]This is because we add the amount of Iron to the number of Iron from the vitamins which are 4x (4 is the amount of mg of Irons per vitamin).
So now we solve for x:
[tex]\begin{gathered} 11-0.9\leq4x<22-0.9 \\ 10.1\leq4x\leq21.1 \\ \frac{10.1}{4}\leq x\leq\frac{21.1}{4} \\ 2.52\leq x\leq5.27 \end{gathered}[/tex]We round the lower limit up, to find the closest whole number to 2.52, so we round it to 3.
And we round the upper limit down, from 5.27 to 5.
So, we get:
[tex]3\leq x\leq5[/tex]Least number of whole vitamins: 3
Greater number of whole vitamins: 5
Part B.
Each week has 7 days, so let's calculate how much he spends only on bananas each week.
He eats 3 bananas per day and pays $2 for every 3 bananas, so he pays $2 per day. Thus, in 7 days he pays for the bananas:
[tex]7\times2=14[/tex]$14 just for the bananas each week.
Now, we need to find how much he spends on vitamins. Again, the number of vitamins per week will be "x", to this, we need to subtract 10 because the first 10 vitamins are free, and the rest he pays 1 dollar for every 2 vitamins. The equation will be as follows:
[tex]14+1(\frac{x-10}{2})[/tex]14 is the cost for the bananas of the week, the rest of the expression is x minus 10 (because 10 vitamins are free) then we divide by 2 because he only pays 1 dollar for every 2 vitamins, and multiply that by 1 (1 dollar).
We can simplify the expression to the following equation:
[tex]14+\frac{x-10}{2}[/tex]Now we are told that he spent $20 for this week, so we need to solve the following equation for x:
[tex]14+\frac{x-10}{2}=20[/tex]And find x, the number of vitamins per week:
Subtract 14 to both sides
[tex]\begin{gathered} \frac{x-10}{2}=20-14 \\ \frac{x-10}{2}=6 \end{gathered}[/tex]Multiply both sides by 2:
[tex]\begin{gathered} x-10=6\times2 \\ x-10=12 \end{gathered}[/tex]Now add 10 to both sides:
[tex]\begin{gathered} x=12+10 \\ x=22 \end{gathered}[/tex]He takes 22 vitamins per week, we divide by 7 to find the number of vitamins per day:
[tex]\frac{22}{7}=3.14[/tex]Ans, as we find in part A, the minimum per day, was 3 whole vitamins, so he will be able to meet his iron requirement
