Question 1182635: In a meadow filled with wildflowers, a hiker observes 450 butterflies. Scientists have been studying this meadow for years and have discovered that the number of butterflies depends on the number of flowers in the field. The line of best-fit that they calculated is shown below. If the hiker's observation is analyzed, the residual for the number of butterflies is -21. How many flowers are there in the field?
Number of butterflies= 86+0.25(number of flowers)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula is:
number of butterflies = 86 + 0.25 * number of flowers.
the formula should really be more explicit as shown below.
predicted number of butterflies = 86 + .25 * actual number of flowers.
the residual is the actual number of butterflies minus the predicted number of butterflies.
if the actual is less than the predicted, the residual is negative.
if the actual is more than the predicted, the residual is positive.
you are given that the actual number of butterflies is 450 and the residual is -21.
the formula is:
residual = actual number of butterflies minus the predicted number of butterflies.
this becomes:
-21 = 450 minus the predicted number of butterflies.
add 21 to both sides of the equation and add the predicted number of butterflies to both sides of the equation to get:
predicted number of butterflies = 450 = 21 = 471.
your predicted number of butterflies is 471.
your original formula (after my revision) is:
predicted number of butterflies = 86 + .25 * actual number of flowers.
when the predicted number of butterflieis is 471, the equation becomes:
471 = 86 + .25 * actual number of flower.
subtract 86 from both sides of this equation to get:
471 - 86 = .25 * actual number of flowers.
divide both sides of this equation by .25 to get:
(471 - 86) / .25 = actual number of flowers.
solve for actual number of flowers to get:
actual number of flowers = (471 - 86) / .25 = 1540.
that should be your answer.
confirm by replacing the actual number of flowers in the original equation by 1540 and solving for the predicted number of butterflies.
the original formula is, once again:
predicted number of butterflies = 86 + .25 * actual number of flowers.
when the actual number of flowers is 1540, the formula becomes:
predicted number of butterflies = 86 + .25 * 1540 = 471.
the residual is -21.
this means the actual number of butterflies is 21 less than the predicted number of butterflies.
the formula for residual is:
residual = actual number of butterflies minus predicted number of butterflies.
the formula becomes:
-21 = 450 - predicted number of butterflies.
add 21 to both sides of this equation and add predicted number of butterflies to both sides of this equation to get:
predicted number of butterflies = 450 + 21 = 471.
this confirms the solution is correct, as best i can determine.
the solution is that the number of actual flowers is 1540.
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