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QuestionA handyman company wants to place an office between Jonesville and Bellevue, 60 miles apart. With weights of 3 for Jonesville and 5 for Bellevue, how far from Bellevue should the office be? d=53+5×60d = \frac{5}{3+5} \times 60 miles.

Studdy Solution

STEP 1

Assumptions1. The distance between Jonesville and Bellevue is60 miles. . The weight assigned to Jonesville is3.
3. The weight assigned to Bellevue is5.
4. The location of the new office is determined by the weighted average of the distances to the two cities.

STEP 2

First, we need to understand the concept of a weighted average. In this case, the weights are the number of calls from each city. The weighted average distance, D, from Bellevue can be calculated as follows=WeightJonesville×DistanceJonesville+WeightBellevue×DistanceBellevueWeightJonesville+WeightBellevue = \frac{Weight_{Jonesville} \times Distance_{Jonesville} + Weight_{Bellevue} \times Distance_{Bellevue}}{Weight_{Jonesville} + Weight_{Bellevue}}

STEP 3

Now, we know that the distance from Bellevue to the new office plus the distance from the new office to Jonesville equals the total distance between Bellevue and Jonesville. Let's denote the distance from Bellevue to the new office as x. Then the distance from the new office to Jonesville is (60 - x). We can rewrite the weighted average formula as follows=WeightJonesville×(60−x)+WeightBellevue×xWeightJonesville+WeightBellevue = \frac{Weight_{Jonesville} \times (60 - x) + Weight_{Bellevue} \times x}{Weight_{Jonesville} + Weight_{Bellevue}}

STEP 4

We know that the weighted average distance, D, is the same as the distance from Bellevue to the new office, which is x. So, we can set D equal to x and solve for xx=WeightJonesville×(60−x)+WeightBellevue×xWeightJonesville+WeightBellevuex = \frac{Weight_{Jonesville} \times (60 - x) + Weight_{Bellevue} \times x}{Weight_{Jonesville} + Weight_{Bellevue}}

STEP 5

Now, plug in the given values for the weights of Jonesville and Bellevue to solve for xx=3×(60−x)+5×x3+5x = \frac{3 \times (60 - x) +5 \times x}{3 +5}

STEP 6

implify the equationx=180−3x+5x8x = \frac{180 -3x +5x}{8}

STEP 7

Combine like termsx=180+2xx = \frac{180 +2x}{}

STEP 8

Multiply both sides of the equation by8 to get rid of the denominator on the right side8x=180+2x8x =180 +2x

STEP 9

Subtract2x from both sides of the equation6x=1806x =180

STEP 10

Divide both sides of the equation by6 to solve for xx=1806x = \frac{180}{6}

STEP 11

Calculate the value of xx=1806=30x = \frac{180}{6} =30The new office should be located30 miles from Bellevue.

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