Radical Equations

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Radical Equations · Grade 9 · Worksheet 2

1. A marine biologist is studying the relationship between water depth and light intensity in a coral reef. The light intensity I (in lumens) at depth d (in meters) is given by the formula I = 120/√(d + 4). If the light intensity measures 24 lumens at a certain depth, what is the depth in meters? Answer: 21 Solution: Start with the given formula: I = 120/√(d + 4) Substitute I = 24 into the equation: 24 = 120/√(d + 4) Multiply both sides by √(d + 4): 24√(d + 4) = 120 Divide both sides by 24: √(d + 4) = 120/24 Simplify: √(d + 4) = 5 Square both sides: d + 4 = 25 Subtract 4 from both sides: d = 21 The depth is…
   Full step-by-step solution

   Step 1: Start with the given formula: I = 120/√(d + 4) Step 2: Substitute I = 24 into the equation: 24 = 120/√(d + 4) Step 3: Multiply both sides by √(d + 4): 24√(d + 4) = 120 Step 4: Divide both sides by 24: √(d + 4) = 120/24 Step 5: Simplify: √(d + 4) = 5 Step 6: Square both sides: d + 4 = 25 Step 7: Subtract 4 from both sides: d = 21 The depth is 21 meters.

2. Charlotte is a physicist studying the motion of a pendulum. The time T (in seconds) for one complete swing is modeled by the equation T = 2π√(L/9.8), where L is the length of the pendulum in meters. In a specific experiment, she measures the time for one swing as 2π seconds. What is the length of the pendulum in meters? Answer: 9.8 Solution: Start with the equation T = 2π√(L/9.8). Substitute T = 2π: 2π = 2π√(L/9.8). Divide both sides by 2π to isolate the square root: 1 = √(L/9.8).
   Full step-by-step solution

   Step 1: Start with the equation T = 2π√(L/9.8). Substitute T = 2π: 2π = 2π√(L/9.8). Step 2: Divide both sides by 2π to isolate the square root: 1 = √(L/9.8). Step 3: Square both sides to eliminate the square root: 1² = (√(L/9.8))² → 1 = L/9.8. Step 4: Multiply both sides by 9.8: L = 9.8. Step 5: Check: 2π√(9.8/9.8) = 2π√1 = 2π(1) = 2π. The solution is valid. The length of the pendulum is 9.8 meters.

3. Emma is a materials scientist testing a new alloy. The length L (in centimeters) of a metal rod under tension follows the equation L = sqrt(5x + 75), where x is the force applied in Newtons. If the rod's length is measured as 15 centimeters, what force x (in Newtons) is being applied? Answer: 30 Solution: Write the equation with the given length: sqrt(5x + 75) = 15 Square both sides to eliminate the square root: (sqrt(5x + 75))^2 = 15^2 Simplify: 5x + 75 = 225 Subtract 75 from both sides: 5x = 150 Divide both sides by 5: x = 30 Check by substituting x = 30 back into the original equation:…
   Full step-by-step solution

   Step 1: Write the equation with the given length: sqrt(5x + 75) = 15 Step 2: Square both sides to eliminate the square root: (sqrt(5x + 75))^2 = 15^2 Step 3: Simplify: 5x + 75 = 225 Step 4: Subtract 75 from both sides: 5x = 150 Step 5: Divide both sides by 5: x = 30 Step 6: Check by substituting x = 30 back into the original equation: sqrt(5(30) + 75) = sqrt(150 + 75) = sqrt(225) = 15. The solution is valid. The force applied is 30 Newtons.

4. Aroha is an aerospace engineer designing a parabolic reflector for a satellite dish. The depth of the dish, in centimeters, at a horizontal distance x centimeters from the center is given by the formula d(x) = sqrt(2x + 65). For optimal signal reception, the depth at the edge of the dish must be exactly 13 centimeters. What is the horizontal distance from the center to the edge of the dish in centimeters? Answer: 52 Solution: Set up the equation using the given depth: sqrt(2x + 65) = 13 Square both sides to eliminate the square root: (sqrt(2x + 65))^2 = 13^2 Simplify: 2x + 65 = 169 Subtract 65 from both sides: 2x = 104 Divide both sides by 2: x = 52 Check the solution by plugging x = 52 back into the original…
   Full step-by-step solution

   Step 1: Set up the equation using the given depth: sqrt(2x + 65) = 13 Step 2: Square both sides to eliminate the square root: (sqrt(2x + 65))^2 = 13^2 Step 3: Simplify: 2x + 65 = 169 Step 4: Subtract 65 from both sides: 2x = 104 Step 5: Divide both sides by 2: x = 52 Step 6: Check the solution by plugging x = 52 back into the original equation: sqrt(2(52) + 65) = sqrt(104 + 65) = sqrt(169) = 13. The solution is valid. The horizontal distance from the center to the edge of the dish is 52 centimeters.

5. Isabella is a materials scientist testing the strength of a new alloy beam. The deflection d (in millimeters) of the beam under a load is modeled by the equation d = √(4x + 28), where x is the distance in meters from the fixed support. For a safety test, she needs the deflection to be exactly 8 millimeters. At what distance x from the fixed support does this deflection occur? Answer: 9 Solution: Set up the equation using the given deflection: √(4x + 28) = 8 Square both sides to eliminate the square root: (√(4x + 28))^2 = 8^2 Simplify: 4x + 28 = 64 Subtract 28 from both sides: 4x = 36 Divide both sides by 4: x = 9 Check the solution by plugging x = 9 back into the original equation:…
   Full step-by-step solution

   Step 1: Set up the equation using the given deflection: √(4x + 28) = 8 Step 2: Square both sides to eliminate the square root: (√(4x + 28))^2 = 8^2 Step 3: Simplify: 4x + 28 = 64 Step 4: Subtract 28 from both sides: 4x = 36 Step 5: Divide both sides by 4: x = 9 Step 6: Check the solution by plugging x = 9 back into the original equation: √(4(9) + 28) = √(36 + 28) = √64 = 8. The solution checks out. The distance x from the fixed support is 9 meters.

6. Sophia is a safety inspector testing the braking distance of a new electric vehicle. The braking distance d (in meters) on a dry road is given by the formula d = √(0.5v + 13), where v is the speed of the car in kilometers per hour (km/h) just before braking. During a test, the vehicle's braking distance is measured as exactly 7 meters. What was the speed of the car just before braking, in km/h? Answer: 72 Solution: Write the equation using the given braking distance: √(0.5v + 13) = 7. Square both sides to eliminate the square root: (√(0.5v + 13))² = 7² → 0.5v + 13 = 49. Subtract 13 from both sides: 0.5v = 49 - 13 → 0.5v = 36.
   Full step-by-step solution

   Step 1: Write the equation using the given braking distance: √(0.5v + 13) = 7. Step 2: Square both sides to eliminate the square root: (√(0.5v + 13))² = 7² → 0.5v + 13 = 49. Step 3: Subtract 13 from both sides: 0.5v = 49 - 13 → 0.5v = 36. Step 4: Divide both sides by 0.5: v = 36 / 0.5 → v = 72. Step 5: Check by substituting v = 72 into the original equation: √(0.5(72) + 13) = √(36 + 13) = √49 = 7. The solution is valid. The speed of the car just before braking was 72 km/h.

7. √(5x - 11) + 2 = 9 Answer: 12 Solution: Subtract 2 from both sides: √(5x - 11) + 2 - 2 = 9 - 2 → √(5x - 11) = 7 Square both sides: (√(5x - 11))² = 7² → 5x - 11 = 49 Add 11 to both sides: 5x - 11 + 11 = 49 + 11 → 5x = 60 Divide both sides by 5: 5x/5 = 60/5 → x = 12 Check solution: √(5(12) - 11) + 2 = √(60 - 11) + 2 = √49 + 2 = 7 + 2 =…
   Full step-by-step solution

   Step 1: Subtract 2 from both sides: √(5x - 11) + 2 - 2 = 9 - 2 → √(5x - 11) = 7 Step 2: Square both sides: (√(5x - 11))² = 7² → 5x - 11 = 49 Step 3: Add 11 to both sides: 5x - 11 + 11 = 49 + 11 → 5x = 60 Step 4: Divide both sides by 5: 5x/5 = 60/5 → x = 12 Step 5: Check solution: √(5(12) - 11) + 2 = √(60 - 11) + 2 = √49 + 2 = 7 + 2 = 9 ✓ The answer is 12.