# Find the exact length of the curve x = 1/3 √y (y − 3), 16 ≤ y ≤ 25.

We are going to use using the formula of the exact length of the curve to solve this.

## Answer: The exact length of the curve x = 1/3 √y (y − 3), 16 ≤ y ≤ 25 is 64/3.

Let's solve this step by step.

**Explanation:**

Given, x = 1/3 √y (y − 3), 16 ≤ y ≤ 25

Length of the curve y = f(x) from x =a to x = b is given by: \(\int\limits_a^b\) √1 + [f′(x)]^{2} dx.

x = 1/3 √y (y − 3)

x = 1/3 . y^{3/2 }- y^{1/2}

Let's find the first derivative of x.

dx/dy = 1/3. 3/2. y^{1/2}- 1/2 . y^{1/2}

dx/ dy = 1/2 ( y^{1/2 }-y^{-1/2})

(dx/dy)^{2} = 1/4 ( y^{1/2 }-y^{-1/2})^{2}

= 1/4(y + y^{-1 }-2)

1 + {f′(x)}^{2 }= 1 + 1/4(y + y^{-1 }-2)

= 1/4 y + 1/4y + 1/2

1 + {f′(x)}^{2}= 1/4 (y + y^{-1} + 2)

√[1 + {f′(x)}^{2}] = 1/2 ( y^{1/2 }+ y^{-1/2})

Length of curve = \(\int\limits_{16}^{25}\) √[1 + {f′(x)}^{2}] dx

= \(\int\limits_{16}^{25}\) √y/2 + 1/2√y dy

= [(y)^{3/2 }/3 + √y]\(^{25}_{16}\)

= [(25)^{3/2 }/3 + √25] - [(16)^{3/2 }/3 + √16]

= [125/3 + 5] - [64/3 + 4]

= 140/3 - 76/3

= (140 - 76)/3

= 64/3

### Hence, the exact length of the curve x = 1/3 √y (y − 3), 16 ≤ y ≤ 25 is 64/3.

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