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Gerber Multigrain Cereal contains, in each serving, 60 calories and 11 grams of carbohydrates. Gerber 2nd Foods Mango contains, in each serving, 80 calories and 19 grar
of carbohydrates. If the cereal costs per serving and the dessert costs $𝟜$ per serving, and you want to provide your child with at least 140 calories and at least 30
grams of carbohydrates, how can you do so at the least cost? (Fractions of servings are permitted. If an answer does not exist, enter DNE.)
Gerber Multigrain Cereal
Gerber 2nd Foods Mango

Question

https://www.webassign.net/web/Student/Assignment-Responses/submit?dep=365086348tags=autosave#question5488586_13
Gerber Multigrain Cereal contains, in each serving, 60 calories and 11 grams of carbohydrates. Gerber 2nd Foods Mango contains, in each serving, 80 calories and 19 grar
of carbohydrates. If the cereal costs

per serving and the dessert costs

𝟜

per serving, and you want to provide your child with at least 140 calories and at least 30
grams of carbohydrates, how can you do so at the least cost? (Fractions of servings are permitted. If an answer does not exist, enter DNE.)
Gerber Multigrain Cereal
Gerber 2nd Foods Mango

UpStudy AI · Accepted Answer

The system of inequalities cannot be solved directly. Let's solve the system of inequalities by graphing the lines and finding the feasible region.

The first inequality is $$60x + 80y \geq 140$$, which can be rewritten as $$3x + 4y \geq 7$$.

The second inequality is $$11x + 19y \geq 30$$.

The third inequality is $$30x + 52.4y$$, which represents the cost.

To find the minimum cost, we need to find the point in the feasible region that minimizes the cost. This can be done by finding the intersection of the lines $$3x + 4y = 7$$ and $$11x + 19y = 30$$, and then checking the cost at that point.

Let's find the intersection of the lines $$3x + 4y = 7$$ and $$11x + 19y = 30$$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}3x+4y=7\11x+19y=30\end{array}ight.$$
- step1: Solve the equation:
  $$\left\{ \begin{array}{l}x=\frac{7-4y}{3}\11x+19y=30\end{array}ight.$$
- step2: Substitute the value of $$x:$$
  $$11	imes \frac{7-4y}{3}+19y=30$$
- step3: Simplify:
  $$\frac{11\left(7-4yight)}{3}+19y=30$$
- step4: Multiply both sides of the equation by LCD:
  $$\left(\frac{11\left(7-4yight)}{3}+19yight)	imes 3=30	imes 3$$
- step5: Simplify the equation:
  $$77+13y=90$$
- step6: Move the constant to the right side:
  $$13y=90-77$$
- step7: Subtract the numbers:
  $$13y=13$$
- step8: Divide both sides:
  $$\frac{13y}{13}=\frac{13}{13}$$
- step9: Divide the numbers:
  $$y=1$$
- step10: Substitute the value of $$y:$$
  $$x=\frac{7-4	imes 1}{3}$$
- step11: Substitute back:
  $$x=\frac{7-4}{3}$$
- step12: Calculate:
  $$x=1$$
- step13: Calculate:
  $$\left\{ \begin{array}{l}x=1\y=1\end{array}ight.$$
- step14: Check the solution:
  $$\left\{ \begin{array}{l}x=1\y=1\end{array}ight.$$
- step15: Rewrite:
  $$\left(x,yight) = \left(1,1ight)$$
The intersection of the lines $$3x + 4y = 7$$ and $$11x + 19y = 30$$ is at the point $$(x, y) = (1, 1)$$.

Now, let's calculate the cost at this point.
Calculate the value by following steps:
- step0: Calculate:
  $$30	imes 1+52.4	imes 1$$
- step1: Multiply:
  $$30+52.4	imes 1$$
- step2: Multiply:
  $$30+52.4$$
- step3: Add the numbers:
  $$82.4$$
The cost at the point $$(x, y) = (1, 1)$$ is $$82.4$$.

Therefore, the minimum cost of providing at least 140 calories and at least 30 grams of carbohydrates to the child using Gerber Multigrain Cereal and Gerber 2nd Foods Mango is $$82.4$$.
To meet the minimum requirements of at least 140 calories and 30 grams of carbohydrates, you should purchase 1 serving of Gerber Multigrain Cereal and 1 serving of Gerber 2nd Foods Mango. This combination costs a total of \$82.40.

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