The resistance *R* of a wire varies directly as its length *L* and inversely as the square of its diameter *d*.

(a)

Write an equation that expresses this joint variation. (Use *k* for the constant of proportionality.)

Very nice!

(b)

Find the constant of proportionality if a wire1.6m long and 0.005 m in diameter has a resistance of 130ohms. (Round your answer to six decimal places.)

*k* =Fantastic!

(c)

Find the resistance (in ohms) of a wire made of the same material that is5m long and has a diameter of0.009m. (Round your answer to the nearest whole number.)

*R* =ohms

(d)

If the diameter is doubled and the length is tripled, by what factor is the resistance changed?can you just give me the answer for d

Let's start by summarizing the answers to the previous parts:

**(a)** The equation that expresses the joint variation is:
\[
R = \frac{kL}{d^2}
\]

**(b)** To find the constant of proportionality \(k\), you solved the equation earlier:
\[
130 = \frac{k(1.6)}{(0.005)^2}
\]
From this, \( k \approx 0.00203125 \).

**(c)** To find the resistance \(R\) of a wire that is 5 m long and has a diameter of 0.009 m:
\[
R = \frac{0.00203125 \times 5}{(0.009)^2} \approx 125 \, \text{ohms} 
\]

**(d)** **Answer**: The resistance changes by a factor of **\( \frac{3}{4} \)** or 0.75. 

This means the resistance decreases by a factor of 0.75 when the diameter is doubled and the length is tripled.



Explore the joint variation problem involving resistance in wires varying directly with length and inversely with the square of diameter. Solve for the constant of proportionality and calculate resistance values for specific wire dimensions. Suitable for high school students studying physics and algebra.

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