Understand the Pricing Model:
The price of the beam depends on the square of its length. Let the original length of the beam be \(L\). Original price = \(L^2\).
Length of Broken Pieces:
The beam breaks into two pieces in the ratio \(4 : 9\). Lengths of the pieces are:
Piece 1: \(\frac{4}{13}L\), Piece 2: \(\frac{9}{13}L\).
Price of the Broken Pieces:
The price of each piece is proportional to the square of its length:
Price of Piece 1: \(\left(\frac{4}{13}L\right)^2 = \frac{16}{169}L^2\)
Price of Piece 2: \(\left(\frac{9}{13}L\right)^2 = \frac{81}{169}L^2\)
Total Price of the Broken Pieces:
\[ \text{Total Price} = \frac{16}{169}L^2 + \frac{81}{169}L^2 = \frac{97}{169}L^2 \]
Loss Calculation:
Original price = \(L^2\). Loss = Original price - Price of broken pieces:
\[ \text{Loss} = L^2 - \frac{97}{169}L^2 = \frac{169}{169}L^2 - \frac{97}{169}L^2 = \frac{72}{169}L^2 \]
Percentage loss:
\[ \text{Percentage Loss} = \frac{\text{Loss}}{\text{Original Price}} \times 100 = \frac{\frac{72}{169}L^2}{L^2} \times 100 = \frac{72}{169} \times 100 = 44.44\%. \]
Thus, the percentage loss is 44.44%.
A furniture trader deals in tables and chairs. He has Rs. 75,000 to invest and a space to store at most 60 items. A table costs him Rs. 1,500 and a chair costs him Rs. 1,000. The trader earns a profit of Rs. 400 and Rs. 250 on a table and chair, respectively. Assuming that he can sell all the items that he can buy, which of the following is/are true for the above problem:
(A) Let the trader buy \( x \) tables and \( y \) chairs. Let \( Z \) denote the total profit. Thus, the mathematical formulation of the given problem is:
\[ Z = 400x + 250y, \]
subject to constraints:
\[ x + y \leq 60, \quad 3x + 2y \leq 150, \quad x \geq 0, \quad y \geq 0. \]
(B) The corner points of the feasible region are (0, 0), (50, 0), (30, 30), and (0, 60).
(C) Maximum profit is Rs. 19,500 when trader purchases 60 chairs only.
(D) Maximum profit is Rs. 20,000 when trader purchases 50 tables only.
Choose the correct answer from the options given below:
A | B | C | D | Average |
---|---|---|---|---|
3 | 4 | 4 | ? | 4 |
3 | ? | 5 | ? | 4 |
? | 3 | 3 | ? | 4 |
? | ? | ? | ? | 4.25 |
4 | 4 | 4 | 4.25 |