Subjects algebra

Parking Revenue E9E5Ff

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1. **State the problem:** We have a matrix of vehicle counts over three days and a charge matrix for each vehicle type. We want to find which revenue statement is true based on the matrix multiplication. 2. **Define the matrices:** Vehicle count matrix $V = \begin{bmatrix}70 & 5 & 7 \\ 81 & 3 & 2 \\ 85 & 2 & 15\end{bmatrix}$ where rows are Thursday, Friday, Saturday and columns are Cars, Buses, Bicycles. Charge matrix $C = \begin{bmatrix}6 \\ 15 \\ 0\end{bmatrix}$ since cars cost 6, buses 15, bicycles 0. 3. **Matrix multiplication for revenue:** Revenue matrix $R = V \times C = \begin{bmatrix}70 & 5 & 7 \\ 81 & 3 & 2 \\ 85 & 2 & 15\end{bmatrix} \times \begin{bmatrix}6 \\ 15 \\ 0\end{bmatrix}$ 4. **Calculate revenue for each day:** - Thursday revenue: $$70 \times 6 + 5 \times 15 + 7 \times 0 = 420 + 75 + 0 = 495$$ - Friday revenue: $$81 \times 6 + 3 \times 15 + 2 \times 0 = 486 + 45 + 0 = 531$$ - Saturday revenue: $$85 \times 6 + 2 \times 15 + 15 \times 0 = 510 + 30 + 0 = 540$$ 5. **Check the statements:** - "Buses over the three days was 531": Total buses revenue = $5 \times 15 + 3 \times 15 + 2 \times 15 = 75 + 45 + 30 = 150$, so false. - "Cars on Thursday was 540": Cars on Thursday revenue = $70 \times 6 = 420$, so false. - "All vehicles on Thursday was 495": True, as calculated. - "All vehicles on Saturday was 531": Saturday total revenue is 540, so false. **Final answer:** The true statement is **All vehicles on Thursday was 495**.