Question

the members of a running club are selling bottles of lemonade and water to raise money for new uniforms. each bottle of lemonade sells for $2.25, and each bottle of water sells for $1.50. the running club has a total of 375 bottles of lemonade and water. let l represent the number of bottles of lemonade and let w represent the number of bottles of water. the club members need to raise at least $625 for the uniforms. which of the following is a true statement? select all that are correct.
\\(\

$$\begin{cases} 2.25l + 1.50w \\geq 625 \\\\ l + w \\leq 375 \\end{cases}$$

\\)
\\(\

$$\begin{cases} 2.25l + 1.50w \\leq 625 \\\\ l + w \\leq 375 \\end{cases}$$

\\)
if the club members sell 150 bottles of water, they would be able sell enough bottles of lemonade to reach their fundraising goal.
if the club members sell 250 bottles of water, they would be able sell enough bottles of lemonade to reach their fundraising goal.
if the club members sell 300 bottles of water, they would be able sell enough bottles of lemonade to reach their fundraising goal.

Explanation:

Step1: Analizar la ecuación de ingresos

El ingreso total es \(2.25l + 1.50w\) y deben recaudar al menos \$625, así que \(2.25l + 1.50w \geq 625\). El número total de botellas es \(l + w\) y no puede exceder 375, así que \(l + w \leq 375\). Entonces el primer sistema es correcto.

Step2: Analizar la venta de 150 botellas de agua

Si \(w = 150\), entonces \(l \leq 375 - 150 = 225\). El ingreso mínimo necesario es \(625 - 1.50\times150 = 625 - 225 = 400\). Entonces \(l \geq \frac{400}{2.25} \approx 177.78\). Como \(l \leq 225\) y \(177.78 \leq 225\), se puede alcanzar la meta.

Step3: Analizar la venta de 250 botellas de agua

Si \(w = 250\), entonces \(l \leq 375 - 250 = 125\). El ingreso mínimo necesario es \(625 - 1.50\times250 = 625 - 375 = 250\). Entonces \(l \geq \frac{250}{2.25} \approx 111.11\). Como \(111.11 \leq 125\), se puede alcanzar la meta.

Step4: Analizar la venta de 300 botellas de agua

Si \(w = 300\), entonces \(l \leq 375 - 300 = 75\). El ingreso mínimo necesario es \(625 - 1.50\times300 = 625 - 450 = 175\). Entonces \(l \geq \frac{175}{2.25} \approx 77.78\). Pero \(l \leq 75\), lo que es menor que 77.78, así que no se puede alcanzar la meta.

Answer:

• \(

$$\begin{cases} 2.25l + 1.50w \geq 625 \\ l + w \leq 375 \end{cases}$$

\)

• If the club members sell 150 bottles of water, they would be able sell enough bottles of lemonade to reach their fundraising goal.
• If the club members sell 250 bottles of water, they would be able sell enough bottles of lemonade to reach their fundraising goal.