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A landscaping company mixes its own proprietary blend of topsoil that costs per ton to make. The blend contains soil that costs per ton and organic compost that costs per ton. If the company already has tons of soil on hand, how many tons of compost do they need to mix in? Write your answer as a whole number or as a decimal rounded to the nearest tenth. tons
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Q. A landscaping company mixes its own proprietary blend of topsoil that costs per ton to make. The blend contains soil that costs per ton and organic compost that costs per ton. If the company already has tons of soil on hand, how many tons of compost do they need to mix in? Write your answer as a whole number or as a decimal rounded to the nearest tenth. tons
- Denote compost needed as : Let's denote the number of tons of compost needed as . The cost of the soil on hand is per ton, and the company has tons of it. The cost of the compost is per ton.
- Calculate cost of soil: The total cost of the soil on hand is tons multiplied by per ton.Total cost of soil = tons *
- Calculate total cost of soil: Calculate the total cost of the soil.Total cost of soil = $\(18\) \times \$\(27\).\(65\)\(\newline\)Total cost of soil = \$\(497\).\(70\)
- Calculate total cost of final blend: The total cost of the final blend for the total weight \(18 \text{ tons} + x \text{ tons}\) should be equal to the sum of the cost of the soil and the cost of the compost.\(\newline\)Total cost of final blend = \(18 + x\) * \(\$29.15\)
- Calculate cost of compost: The cost of the compost that needs to be added is \(x\) tons multiplied by \(\$42.65\) per ton.\(\newline\)Cost of compost = \(x \times \$42.65\)
- Set up equation for total cost: Set up the equation based on the total cost of the final blend being the sum of the cost of the soil and the cost of the compost.\(\$497.70 + (x \times \$42.65) = (18 + x) \times \$29.15\)
- Distribute \(\$29.15\) on right side: Distribute the \(\$29.15\) on the right side of the equation.\(\newline\)\(\$497.70 + (x \times \$42.65) = \$525.70 + (x \times \$29.15)\)
- Subtract \(\$497.70\) to isolate \(x\): Subtract \(\$497.70\) from both sides of the equation to isolate the terms with \(x\) on one side.\(\newline\)\((x \times \$42.65) = \$525.70 + (x \times \$29.15) - \$497.70\)
- Simplify right side of equation: Simplify the right side of the equation. \(x \times \$42.65\) = \$\(28\).\(00\) + \(x \times \$29.15\)
- Subtract \(\$29.15\) to isolate x: Subtract \((x \times \$29.15)\) from both sides to get x by itself.\(\newline\)\((x \times \$42.65) - (x \times \$29.15) = \$28.00\)
- Factor out x from left side: Factor out x from the left side of the equation.\(\newline\)\(x \times (\$42.65 - \$29.15) = \$28.00\)
- Calculate difference in cost per ton: Calculate the difference in cost per ton between the compost and the blend.\(\newline\)\(\$42.65 - \$29.15 = \$13.50\)
- Divide both sides by \(\$13.50\): Divide both sides by \(\$13.50\) to solve for \(x\).\[x = \frac{\$28.00}{\$13.50}\]
- Calculate value of x: Calculate the value of \(x\).\(x = 2.0741\) tons
- Round \(x\) to nearest tenth: Round \(x\) to the nearest tenth.\(x \approx 2.1\) tons
