if the weight of a fish varies jointly as the length and the square of the girth, and if a fish 11 inches long with a girth of 7 inches weighs 0.7lbs, find the weight of a fish 23 inches long with a girth of 18 inches.
To find the weight of a fish 23 inches long with a girth of 18 inches, we can use the concept of joint variation. Joint variation is when a variable depends on two or more variables directly.
In this case, the weight of the fish depends jointly on its length and the square of its girth. Let's denote the weight as W, the length as L, and the girth as G.
According to the given information, we know that the weight of a fish 11 inches long with a girth of 7 inches is 0.7 pounds. Using this information, we can form a proportion to solve for the constant of variation.
We have:
W ∝ L * G^2
0.7 = k * 11 * 7^2
0.7 = k * 11 * 49
To find the constant of variation (k), we divide both sides of the equation by (11 * 49):
k = 0.7 / (11 * 49)
k ≈ 0.0013 (rounded to four decimal places)
Now that we have the constant of variation, we can use it to find the weight of a fish 23 inches long with a girth of 18 inches.
W = k * L * G^2
W = 0.0013 * 23 * 18^2
W ≈ 7.8846 (rounded to four decimal places)
Therefore, the weight of a fish 23 inches long with a girth of 18 inches is approximately 7.8846 pounds.