The manager at a health foods store is mixing a 75-liter batch of his superfruit juice cocktail.
The cocktail, which costs $34.05 per liter to make, combines mixed fruit juice costing $4.30
per liter and acai berry juice costing $46.80 per liter. How many liters of each should he mix
together to make the cocktail?
Write your answers as whole numbers or as decimals rounded to the nearest tenth.
liters of mixed fruit juice
liters of acai berry juice

Mixed fruit juice costs 4.30 per liter, and the manager needs [tex] x [/tex] liters, so the cost for the mixed fruit juice is [tex] 4.30x [/tex].
Acai berry juice costs 46.80 per liter, and the manager needs [tex] y [/tex] liters, so the cost for the acai berry juice is [tex] 46.80y [/tex].
The total cost of the mixture is $m34.05 per liter, and the total batch is 75 liters, so the total cost is [tex] 34.05 \times 75 = 2553.75 [/tex].

Therefore, we have the equation:

[tex] 4.30x + 46.80y = 2553.75 [/tex]

Now, we have a system of two equations:

[tex] \begin{cases} x + y = 75\textsf{....... Equation 1 } \\ 4.30x + 46.80y = 2553.75\textsf{....... Equation 2 } \end{cases} [/tex]

We can solve this system of equations to find [tex] x [/tex] and [tex] y [/tex].

To simplify the calculations, we'll multiply the first equation by 4.30 to eliminate decimals:

[tex] \begin{cases} 4.30x + 4.30y = 322.5 \\ 4.30x + 46.80y = 2553.75 \end{cases} [/tex]

Now, we'll subtract the first equation from the second to eliminate [tex] x [/tex]:

[tex] (4.30x + 46.80y) - (4.30x + 4.30y) = 2553.75 - 322.5 [/tex]

[tex] 42.50y = 2231.25 [/tex]

[tex] y = \dfrac{2231.25}{42.50} [/tex]

[tex] y \approx 52.5 [/tex]

Now that we have [tex] y [/tex], we can find [tex] x [/tex]:

[tex] x = 75 - y [/tex]

[tex] x \approx 75 - 52.5 [/tex]

[tex] x \approx 22.5 [/tex]

Therefore, the manager should mix approximately 22.5 liters of mixed fruit juice and 52.5 liters of acai berry juice.