# A School is Preparing a Trip for 400 Students

A school is preparing a trip for 400 students. The company providing the transportation has 10 buses with 50 seats each and 8 buses with 40 seats each, but only has 9 drivers available. The rental cost for a large bus is $800 and$600 for a small bus. Calculate how many buses of each type should be used for the trip to minimize the total cost.

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1. To transport 400 students at the least possible cost, the school should use 4 large buses and 5 small buses. The total cost will be \$5200.

2. ### Case-by-Case Analysis

1. $x=9,y=0x = 9, y = 0$x=9,y=0

$50\left(9\right)+40\left(0\right)=450\phantom{\rule{0ex}{0ex}}\left(\ge 400\right)\phantom{\rule{0ex}{0ex}}OK50\left(9\right) + 40\left(0\right) = 450 \quad \left(\geq 400\right) \quad \text\left\{OK\right\}$
$Cost=800\left(9\right)+600\left(0\right)=7200\text\left\{Cost\right\} = 800\left(9\right) + 600\left(0\right) = 7200$

2. $x=8,y=1x = 8, y = 1$x=8,y=1

$50\left(8\right)+40\left(1\right)=440\phantom{\rule{0ex}{0ex}}\left(\ge 400\right)\phantom{\rule{0ex}{0ex}}OK50\left(8\right) + 40\left(1\right) = 440 \quad \left(\geq 400\right) \quad \text\left\{OK\right\}$
$Cost=800\left(8\right)+600\left(1\right)=6800\text\left\{Cost\right\} = 800\left(8\right) + 600\left(1\right) = 6800$

3. $x=7,y=2x = 7, y = 2$x=7,y=2

$50\left(7\right)+40\left(2\right)=430\phantom{\rule{0ex}{0ex}}\left(\ge 400\right)\phantom{\rule{0ex}{0ex}}OK50\left(7\right) + 40\left(2\right) = 430 \quad \left(\geq 400\right) \quad \text\left\{OK\right\}$
$Cost=800\left(7\right)+600\left(2\right)=6400\text\left\{Cost\right\} = 800\left(7\right) + 600\left(2\right) = 6400$

4. $x=6,y=3x = 6, y = 3$x=6,y=3

$50\left(6\right)+40\left(3\right)=420\phantom{\rule{0ex}{0ex}}\left(\ge 400\right)\phantom{\rule{0ex}{0ex}}OK50\left(6\right) + 40\left(3\right) = 420 \quad \left(\geq 400\right) \quad \text\left\{OK\right\}$
$Cost=800\left(6\right)+600\left(3\right)=6000\text\left\{Cost\right\} = 800\left(6\right) + 600\left(3\right) = 6000$

5. $x=5,y=4x = 5, y = 4$x=5,y=4

$50\left(5\right)+40\left(4\right)=410\phantom{\rule{0ex}{0ex}}\left(\ge 400\right)\phantom{\rule{0ex}{0ex}}OK50\left(5\right) + 40\left(4\right) = 410 \quad \left(\geq 400\right) \quad \text\left\{OK\right\}$
$Cost=800\left(5\right)+600\left(4\right)=5600\text\left\{Cost\right\} = 800\left(5\right) + 600\left(4\right) = 5600$

6. $x=4,y=5x = 4, y = 5$x=4,y=5

$50\left(4\right)+40\left(5\right)=400\phantom{\rule{0ex}{0ex}}\left(\ge 400\right)\phantom{\rule{0ex}{0ex}}OK50\left(4\right) + 40\left(5\right) = 400 \quad \left(\geq 400\right) \quad \text\left\{OK\right\}$
$Cost=800\left(4\right)+600\left(5\right)=5200\text\left\{Cost\right\} = 800\left(4\right) + 600\left(5\right) = 5200$