A basketball team is selling merchandise at halftime of the championship game. The hats cost $25, and the jerseys cost $45. The team sells at least 20 jerseys and over double the number of hats as jerseys. They make less than $3,000. If x represents the number of hats sold, and y represents the number of jerseys sold, could the point (62,25)

Question

(
62
,
25
)
represent a viable solution?(1 point)
Responses

No, because it does not lie within the overlapping shaded region and does not make sense in the context of the problem.
No, because it does not lie within the overlapping shaded region and does not make sense in the context of the problem.

Yes, because it lies within the overlapping shaded region, but does not make sense in the context of the problem.
Yes, because it lies within the overlapping shaded region, but does not make sense in the context of the problem.

Yes, because it lies within the overlapping shaded region and makes sense in the context of the problem.
Y

Answer 1

To determine if the point (62, 25) could be a viable solution for the basketball team's merchandise sales, we need to check the given conditions.

1. Cost of hats and jerseys:

   • Each hat costs $25, and each jersey costs $45.
   • If \( x \) is the number of hats sold and \( y \) is the number of jerseys sold, the revenue can be calculated as: \[ 25x + 45y \]

2. Conditions:

   • At least 20 jerseys sold: \( y \geq 20 \)
   • Over double the number of hats as jerseys: \( x > 2y \)
   • Total revenue must be less than $3,000: \( 25x + 45y < 3000 \)

3. Substituting the point (62, 25):

   • Let \( x = 62 \) and \( y = 25 \).

   • Check the jersey condition:
     \[ y = 25 \geq 20 \quad \text{(satisfied)} \]

   • Check the hats condition:
     \[ x = 62 > 2y \quad \Rightarrow \quad 62 > 2(25) \quad \Rightarrow \quad 62 > 50 \quad \text{(satisfied)} \]

   • Check the revenue condition:
     \[ 25x + 45y = 25(62) + 45(25) \] \[ = 1550 + 1125 = 2675 \] \[ 2675 < 3000 \quad \text{(satisfied)} \]

Since all conditions are satisfied, the point (62, 25) does lie within the feasible region and makes sense in context.

Therefore, the correct option is:

Yes, because it lies within the overlapping shaded region and makes sense in the context of the problem.