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.
ScanSolve AI Answer
a) To construct an equilateral triangle of side 5 cm: Step 1: Draw a line segment PQ of length 5 cm using a ruler. Step 2: Place the compass point at P and open it to a radius of 5 cm. Draw an arc above the line segment PQ. Step 3: Place the compass point at Q and, with the same radius of 5 cm, draw another arc that intersects the first arc. Label the intersection point R. Step 4: Draw straight lines from P to R and from Q to R using a ruler. Triangle PQR is the equilateral triangle with sides of 5 cm. b) To construct the circumscribed circle of triangle PQR: Step 1: Construct the perpendicular bisector of side PQ. To do this, place the compass point at P, open it to more than half the length of PQ, and draw arcs above and below PQ. Repeat this from point Q with the same compass opening. Draw a straight line connecting the two points where the arcs intersect. This is the perpendicular bisector of PQ. Step 2: Construct the perpendicular bisector of side QR using the same method as in Step 1. Step 3: The point where the two perpendicular bisectors intersect is the circumcenter of the triangle. Label this point O. Step 4: Place the compass point at O and open it to the distance OP (or OQ or OR, as these distances will be equal). Draw a circle. This circle passes through all three vertices P, Q, and R, and is the circumscribed circle. 20. To calculate the total surface area of a closed cylinder, use the formula: A = 2 r^2 + 2 rh where r is the radius and h is the height. Step 1: Identify the given values. Diameter d = 7 cm Height h = 20 cm = (22)/(7) Step 2: Calculate the radius r. The radius is half of the diameter: r = (d)/(2) = (7)/(2) cm Step 3: Substitute the values into the surface area formula. A = 2 ((22)/(7)) ((7)/(2))^2 + 2 ((22)/(7)) ((7)/(2)) (20) Step 4: Calculate the area of the two circular bases (2 r^2). 2 ((22)/(7)) ((49)/(4)) = 2 × (22)/(7) × (49)/(4) = (2 × 22 × 49)/(7 × 4) = (2 × 22 × 7 × 7)/(7 × 2 × 2) = (22 × 7)/(2) = 11 × 7 = 77 cm^2 Step 5: Calculate the lateral surface area (2 rh). 2 ((22)/(7)) ((7)/(2)) (20) = (2 × 22 × 7 × 20)/(7 × 2) = 22 × 20 = 440 cm^2 Step 6: Add the areas to find the total surface area. A = 77 cm^2 + 440 cm^2 A = 517 cm^2 The total surface area of the cylinder is 517 cm^2. 3 done, 2 left today. You're making progress.