This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Question 4: i) Find the relation connecting R, L and r. Step 1: Write the proportionality statement based on the given information. The resistance R varies directly as its length L, so R L. The resistance R varies inversely as the square of its radius r, so R (1)/(r^2). Combining these, we get: R (L)/(r^2) Step 2: Introduce the constant of proportionality, k. R = (kL)/(r^2) Step 3: Use the given values to find k. Given: R = 30 ohms, L = 500 m, r = 0.2 cm. 30 = (k × 500)/((0.2)^2) 30 = (500k)/(0.04) 30 × 0.04 = 500k 1.2 = 500k k = (1.2)/(500) = (12)/(5000) = (3)/(1250) Step 4: Write the final relation. The relation connecting R, L and r is: R = (3L)/(1250r^2) ii) Calculate the resistance of the same type of copper wire 750 m long and radius 0.25 cm. Step 1: Use the relation found in part i) with the new values. Given: L = 750 m, r = 0.25 cm, and k = (3)/(1250). R = (kL)/(r^2) R = (3)/(1250) × 750(0.25)^2 Step 2: Simplify the expression. R = (3 × 750)/(1250 × (0.25)^2) R = (2250)/(1250 × 0.0625) R = (2250)/(78.125) Alternatively, simplify the fraction: R = (3 × 750)/(1250 × (1/4)^2) R = (3 × 3 × 250)/(5 × 250 × (1/16)) R = (9)/(5 × (1/16)) R = (9)/(5/16) R = (9 × 16)/(5) R = (144)/(5) R = 28.8 The resistance of the wire is: 28.8 ohms Question 5: a) Use the distributive property to expand and simplify (105)/(32) ( (320)/(21) + (864)/(15) ). Step 1: Apply the distributive property. (105)/(32) × (320)/(21) + (105)/(32) × (864)/(15) Step 2: Simplify the first term. (105)/(32) × (320)/(21) = (105)/(21) × (320)/(32) = 5 × 10 = 50 Step 3: Simplify the second term. (105)/(32) × (864)/(15) = (105)/(15) × (864)/(32) = 7 × 27 = 189 Step 4: Add the simplified terms. 50 + 189 = 239 The simplified expression is: 239 b) Without using calculators, simplify 2^3 × 3^2 × 9^-2 × 16^(1)/(2), leaving your answer as a fraction in its lowest term. Step 1: Express all numbers as powers of prime bases. 2^3 × 3^2 × (3^2)^-2 × (2^4)^(1)/(2) Step 2: Apply exponent rules (a^m)^n = a^mn and (a^m)^(1)/(n) = a^(m)/(n). 2^3 × 3^2 × 3^-4 × 2^4 × (1)/(2) 2^3 × 3^2 × 3^-4 × 2^2 Step 3: Group terms with the same base and apply a^m × a^n = a^m+n. (2^3 × 2^2) × (3^2 × 3^-4) 2^3+2 × 3^2-4 2^5 × 3^-2 Step 4: Evaluate the powers. 32 × (1)/(3^2) 32 × (1)/(9) Step 5: Multiply to get the final fraction. (32)/(9) The simplified expression in its lowest term is: (32)/(9) 3 done, 2 left today. You're making progress.