The resistance R of wire varies directly as its length L and inversely as the square of its diameter d
a.write an equation that express this joint variation (use k for the constant proportionality)
B.find the constant of proportionality if a wire 1.8m long and 0.005m in diameter has a resistance of 190ohms (round your answer to six decimal places)k=?
C.Find the resistance of a wire made of the same material that is 4m long and has a diameter of 0.009m (round your answer to the nearest whole number) R=? In ohms
D.if the diameter is doubled and the length is tripled by what factor is the resistance changed?
a. The equation that expresses this joint variation is: R = k * (L / d^2)
b. To find the constant of proportionality, we can plug in the given values into the equation and solve for k:
190 = k * (1.8 / 0.005^2)
190 = k * (1.8 / 0.000025)
k = 190 / (1.8 / 0.000025)
k ≈ 2,638,888.888889 (rounded to six decimal places)
c. To find the resistance, we can plug in the given values into the equation:
R = k * (4 / 0.009^2)
R = 2,638,888.888889 * (4 / 0.000081)
R = 2,638,888.888889 * 49,382.716049
R ≈ 130,304,249,382.716 (rounded to the nearest whole number)
The resistance is approximately 130,304,249,383 ohms.
d. When the diameter is doubled and the length is tripled, the new equation for resistance becomes:
R' = k * ((3L) / (2d)^2)
The factor by which the resistance is changed can be found by comparing R' to the original resistance R:
(R') / R = (k * ((3L) / (2d)^2)) / (k * (L / d^2))
(R') / R = ((3L) / (2d)^2) / (L / d^2)
(R') / R = (3L * d^2) / (2d)^2
(R') / R = (3L * d^2) / 4d^2
(R') / R = 3L / 4d^2
(R') / R = (3/4) * (L / d^2)
Therefore, the resistance is changed by a factor of 3/4.
Translate the following to spanish:
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a. To express the joint variation between resistance R, length L, and diameter d, we can use the equation:
R = k * (L / d^2)
b. In this case, we are given that the wire is 1.8m long, 0.005m in diameter, and has a resistance of 190ohms. Plugging these values into the equation, we can solve for the constant of proportionality k:
190 = k * (1.8 / 0.005^2)
Simplifying, we have:
190 = k * (1.8 / 0.000025)
Dividing both sides by (1.8 / 0.000025):
k = 190 / (1.8 / 0.000025)
k = 211111.11111111 (rounded to six decimal places)
k ≈ 211111.111
c. Now, we need to find the resistance of a wire that is 4m long and has a diameter of 0.009m. Using the equation with the constant of proportionality k we found earlier, we can calculate the resistance R:
R = k * (L / d^2)
R = 211111.11111 * (4 / 0.009^2)
Simplifying further:
R = 211111.11111 * (4 / 0.000081)
R = 211111.11111 * 4938271.6 (approximately)
Calculating the product:
R ≈ 1.038 × 10^12 ohms
R ≈ 1,038,000,000,000 ohms
R ≈ 1,038,000,000,000 Ω (rounded to the nearest whole number)
d. If the diameter is doubled and the length is tripled, we need to determine the factor by which the resistance changes. Let's assume the original resistance is R1, and the new resistance is R2.
Using the equation for resistance:
R1 = k * (L1 / d1^2)
And with the new dimensions:
R2 = k * (L2 / d2^2)
Dividing the two equations, we get:
R2 / R1 = (k * (L2 / d2^2)) / (k * (L1 / d1^2))
R2 / R1 = (L2 / d2^2) / (L1 / d1^2)
R2 / R1 = (L2 * d1^2) / (L1 * d2^2)
Considering that the length is tripled (L2 = 3L1) and the diameter is doubled (d2 = 2d1), we can substitute these values:
R2 / R1 = (3L1 * d1^2) / (L1 * (2d1)^2)
R2 / R1 = (3L1 * d1^2) / (L1 * 4d1^2)
R2 / R1 = 3 / 4
Therefore, the resistance changes by a factor of 3/4 when the length is tripled and the diameter is doubled.
A. To express the joint variation, we can use the following equation:
R = k * (L / d^2)
Where R is the resistance, L is the length of the wire, d is the diameter of the wire, and k is the constant of proportionality.
B. We can use the given values to find the constant of proportionality (k). We have:
R = 190 ohms
L = 1.8 m
d = 0.005 m
Plugging these values into the equation from part A, we get:
190 = k * (1.8 / 0.005^2)
Simplifying further:
190 = k * (1.8 / 0.000025)
To solve for k, divide both sides of the equation by (1.8 / 0.000025):
k = 190 / (1.8 / 0.000025)
Evaluating this expression, we find k to be approximately 0.528749.
C. To find the resistance (R) of a wire with different length and diameter, we can use the equation from part A:
R = k * (L / d^2)
Given:
L = 4 m
d = 0.009 m
k (from part B) = 0.528749
Plugging in these values, we have:
R = 0.528749 * (4 / 0.009^2)
Simplifying further:
R = 0.528749 * (4 / 0.000081)
To find the resistance (R), divide 4 by 0.000081 and then multiply by 0.528749. The resulting value rounded to the nearest whole number will be the resistance in ohms.
D. If the diameter is doubled and the length is tripled, we can determine the factor by which the resistance changes by comparing the two scenarios.
Let's say the original length is L1, the original diameter is d1, and the original resistance is R1.
According to the joint variation equation, we have:
R1 = k * (L1 / d1^2)
If the diameter is doubled and the length is tripled, we have:
R2 = k * ((3 * L1) / (2 * d1)^2)
To find the factor of change, divide the second equation by the first equation:
(R2 / R1) = (k * ((3 * L1) / (2 * d1)^2)) / (k * (L1 / d1^2))
Simplifying further:
(R2 / R1) = (3 * L1) / (2 * d1)^2) / (L1 / d1^2)
Canceling out the k terms, L1 terms, and d1^2 terms, we get:
(R2 / R1) = (3 * L1 * d1^2) / (2 * d1)^2
Simplifying this expression further, we have:
(R2 / R1) = (3 * d1^2) / (4 * d1^2)
The d1^2 terms cancel out on both sides, leaving us with:
(R2 / R1) = 3 / 4
Therefore, the resistance changes by a factor of 3/4 when the diameter is doubled and the length is tripled.