The interior angle of a regular polygon is 20^ exto more than three times the exterior angle. Find the number of sides of the polygon. 5. Simplify the expression (x2 - 5xy + 6y2)/(18y2 - 2x2).

Question

Asked on April 28, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-04-28T07:50:58.788Z

4. The interior angle of a regular polygon is 20^ more than three times the exterior angle. Find the number of sides of the polygon. Step 1: Define the interior and exterior angles. Let n be the number of sides of the regular polygon. Let I be the interior angle and E be the exterior angle. The relationship between the interior and exterior angles is I + E = 180^. The given condition is I = 3E + 20^. Step 2: Solve for the exterior angle E. Substitute I = 180^ - E into the given condition: 180^ - E = 3E + 20^ 180^ - 20^ = 3E + E 160^ = 4E E = (160^)/(4) E = 40^ Step 3: Find the number of sides n. The formula for the exterior angle of a regular polygon is E = (360^)/(n). Substitute the value of E: 40^ = (360^)/(n) n = (360^)/(40^) n = 9 The number of sides of the polygon is 9. 5. Simplify the expression. (x^2 - 5xy + 6y^2)/(18y^2 - 2x^2) Step 1: Factor the numerator. The numerator is a quadratic expression: x^2 - 5xy + 6y^2 = (x - 2y)(x - 3y) Step 2: Factor the denominator. The denominator is 18y^2 - 2x^2. Factor out the common term 2: 18y^2 - 2x^2 = 2(9y^2 - x^2) Recognize the difference of squares pattern, a^2 - b^2 = (a-b)(a+b), where a=3y and b=x: 2(9y^2 - x^2) = 2(3y - x)(3y + x) To match a term in the numerator, we can rewrite (3y - x) as -(x - 3y): 2(-(x - 3y))(3y + x) = -2(x - 3y)(x + 3y) Step 3: Substitute the factored forms into the expression and simplify. ((x - 2y)(x - 3y))/(-2(x - 3y)(x + 3y)) Cancel the common factor (x - 3y): (x - 2y)/(-2(x + 3y)) This can also be written as: (-(x - 2y))/(2(x + 3y)) = (2y - x)/(2(x + 3y)) The simplified expression is (2y - x)/(2(x + 3y)). 6. Solve for x given. 9^x - 82 × 3^x = -81 Step 1: Rewrite the equation in terms of a common base. Notice that 9^x = (3^2)^x = 3^2x = (3^x)^2. Let y = 3^x. Substitute y into the equation: y^2 - 82y = -81 Step 2: Rearrange into a standard quadratic equation. y^2 - 82y + 81 = 0 Step 3: Solve the quadratic equation for y. Factor the quadratic expression. We need two numbers that multiply to 81 and add to -82. These numbers are -1 and -81. (y - 1)(y - 81) = 0 This gives two possible values for y: y - 1 = 0 y = 1 y - 81 = 0 y = 81 Step 4: Substitute back y = 3^x and solve for x. Case 1: y = 1 3^x = 1 Since 3^0 = 1, then x = 0. Case 2: y = 81 3^x = 81 Since 81 = 3^4, then x = 4. The solutions for x are 0 and 4. 7. A man on top of a tower 300m sees two cars P and Q on a straight level road. The angle of depression of P was 48^ and that of Q was 28^. Calculate the distance between the two cars. (Give your answer to 2.d.p) Step 1: Draw a diagram and identify the relevant right-angled triangles. Let the height of the tower be H = 300 m. Let B be the base of the tower and T be the top. The angles of depression are measured from the horizontal line at T. The angle of depression to P is 48^, so the angle of elevation from P to T is TPB = 48^. The angle of depression to Q is 28^, so the angle of elevation from Q to T is TQB = 28^. We have two right-angled triangles: TBP and TBQ. Step 2: Calculate the horizontal distance from the base of the tower to car P (BP). In TBP: ( TPB) = oppositeadjacent = (TB)/(BP) (48^) = (300)/(BP) BP = (300)/((48^)) BP ≈ (300)/(1.1106125) ≈ 270.1240 m Step 3: Calculate the horizontal distance from the base of the tower to car Q (BQ). In TBQ: ( TQB) = oppositeadjacent = (TB)/(BQ) (28^) = (300)/(BQ) BQ = (300)/((28^)) BQ ≈ (300)/(0.5317094) ≈ 564.2311 m Step 4: Calculate the distance between the two cars (PQ). Since P and Q are on a straight level road and P is closer to the tower than Q (because 48^ > 28^), the distance between the cars is the difference between BQ and BP. PQ = BQ - BP PQ ≈ 564.2311 - 270.1240 PQ ≈ 294.1071 m Step 5: Round the answer to 2 decimal places. PQ ≈ 294.11 m The distance between the two cars is 294.11 m. Send me the next one 📸

Accepted Answer · 2026-04-28T07:50:58.788Z

4. The interior angle of a regular polygon is 20^ more than three times the exterior angle. Find the number of sides of the polygon. Step 1: Define the interior and exterior angles. Let n be the number of sides of the regular polygon. Let I be the interior angle and E be the exterior angle. The relationship between the interior and exterior angles is I + E = 180^. The given condition is I = 3E + 20^. Step 2: Solve for the exterior angle E. Substitute I = 180^ - E into the given condition: 180^ - E = 3E + 20^ 180^ - 20^ = 3E + E 160^ = 4E E = (160^)/(4) E = 40^ Step 3: Find the number of sides n. The formula for the exterior angle of a regular polygon is E = (360^)/(n). Substitute the value of E: 40^ = (360^)/(n) n = (360^)/(40^) n = 9 The number of sides of the polygon is 9. 5. Simplify the expression. (x^2 - 5xy + 6y^2)/(18y^2 - 2x^2) Step 1: Factor the numerator. The numerator is a quadratic expression: x^2 - 5xy + 6y^2 = (x - 2y)(x - 3y) Step 2: Factor the denominator. The denominator is 18y^2 - 2x^2. Factor out the common term 2: 18y^2 - 2x^2 = 2(9y^2 - x^2) Recognize the difference of squares pattern, a^2 - b^2 = (a-b)(a+b), where a=3y and b=x: 2(9y^2 - x^2) = 2(3y - x)(3y + x) To match a term in the numerator, we can rewrite (3y - x) as -(x - 3y): 2(-(x - 3y))(3y + x) = -2(x - 3y)(x + 3y) Step 3: Substitute the factored forms into the expression and simplify. ((x - 2y)(x - 3y))/(-2(x - 3y)(x + 3y)) Cancel the common factor (x - 3y): (x - 2y)/(-2(x + 3y)) This can also be written as: (-(x - 2y))/(2(x + 3y)) = (2y - x)/(2(x + 3y)) The simplified expression is (2y - x)/(2(x + 3y)). 6. Solve for x given. 9^x - 82 × 3^x = -81 Step 1: Rewrite the equation in terms of a common base. Notice that 9^x = (3^2)^x = 3^2x = (3^x)^2. Let y = 3^x. Substitute y into the equation: y^2 - 82y = -81 Step 2: Rearrange into a standard quadratic equation. y^2 - 82y + 81 = 0 Step 3: Solve the quadratic equation for y. Factor the quadratic expression. We need two numbers that multiply to 81 and add to -82. These numbers are -1 and -81. (y - 1)(y - 81) = 0 This gives two possible values for y: y - 1 = 0 y = 1 y - 81 = 0 y = 81 Step 4: Substitute back y = 3^x and solve for x. Case 1: y = 1 3^x = 1 Since 3^0 = 1, then x = 0. Case 2: y = 81 3^x = 81 Since 81 = 3^4, then x = 4. The solutions for x are 0 and 4. 7. A man on top of a tower 300m sees two cars P and Q on a straight level road. The angle of depression of P was 48^ and that of Q was 28^. Calculate the distance between the two cars. (Give your answer to 2.d.p) Step 1: Draw a diagram and identify the relevant right-angled triangles. Let the height of the tower be H = 300 m. Let B be the base of the tower and T be the top. The angles of depression are measured from the horizontal line at T. The angle of depression to P is 48^, so the angle of elevation from P to T is TPB = 48^. The angle of depression to Q is 28^, so the angle of elevation from Q to T is TQB = 28^. We have two right-angled triangles: TBP and TBQ. Step 2: Calculate the horizontal distance from the base of the tower to car P (BP). In TBP: ( TPB) = oppositeadjacent = (TB)/(BP) (48^) = (300)/(BP) BP = (300)/((48^)) BP ≈ (300)/(1.1106125) ≈ 270.1240 m Step 3: Calculate the horizontal distance from the base of the tower to car Q (BQ). In TBQ: ( TQB) = oppositeadjacent = (TB)/(BQ) (28^) = (300)/(BQ) BQ = (300)/((28^)) BQ ≈ (300)/(0.5317094) ≈ 564.2311 m Step 4: Calculate the distance between the two cars (PQ). Since P and Q are on a straight level road and P is closer to the tower than Q (because 48^ > 28^), the distance between the cars is the difference between BQ and BP. PQ = BQ - BP PQ ≈ 564.2311 - 270.1240 PQ ≈ 294.1071 m Step 5: Round the answer to 2 decimal places. PQ ≈ 294.11 m The distance between the two cars is 294.11 m. Send me the next one 📸

The interior angle of a regular polygon is 20^ exto more than three times the exterior angle. Find the number of sides of the polygon. 5. Simplify the expression (x2 - 5xy + 6y2)/(18y2 - 2x2).

Need help with your own homework?

More Mathematics Questions

The interior angle of a regular polygon is 20^ exto more than three times the exterior angle. Find the number of sides of the polygon. 5. Simplify the expression (x2 - 5xy + 6y2)/(18y2 - 2x2).

Need help with your own homework?

More Mathematics Questions