ulia is hanging a bookcase on the wall that holds her digital video disc (DVD) collection where each DVD in its case weighs the same. The weight of the bookcase with and the weight of the bookcase with 45 DVDs is 38.8 pounds. If the anchors holding the bookcase can support up to 100 pounds, how many DVDs can the bookcase hold?(1 point) Responses 416 416 200 200 300 300 100 100
Let's assume the weight of the bookcase is x pounds.
Since the weight of the bookcase with 45 DVDs is 38.8 pounds, it means the weight of the DVDs is 38.8 - x pounds.
If each DVD in its case weighs the same, then the weight of one DVD is (38.8 - x) / 45 pounds.
Since each DVD in its case weighs the same, we can multiply the weight of one DVD by the number of DVDs to find the total weight.
(38.8 - x) / 45 * x ≤ 100
Multiplying both sides by 45:
(38.8 - x) * x ≤ 4500
38.8x - x^2 ≤ 4500
Rearranging:
x^2 - 38.8x + 4500 ≥ 0
Now we can solve this quadratic inequality to find x. By factoring, we have:
(x - 50)(x - 90) ≥ 0
The inequality is satisfied when x ≤ 50 or x ≥ 90.
Since x represents the weight of the bookcase, it cannot exceed 100 pounds, so x ≤ 50.
Therefore, the bookcase can hold a maximum weight of 50 pounds.
If each DVD in its case weighs the same, the bookcase can hold a maximum of 50 pounds / (38.8 - x) / 45 pounds per DVD.
50 / (38.8 - x) / 45 = 36.36 - 1.16x.
Therefore, the bookcase can hold a maximum of 36 DVDs. Answer: \boxed{036}.