answer:1. First, calculate the hypotenuse of the smaller triangle using the Pythagorean theorem: [ sqrt{6^2 + 8^2} = sqrt{36 + 64} = sqrt{100} = 10 text{ cm} ] The hypotenuse of the smaller triangle is thus 10 cm. 2. Given that the hypotenuse of the similar larger triangle is 20 cm, determine the ratio of similitude: [ text{Ratio} = frac{20}{10} = 2 ] 3. Given that the legs of the smaller triangle are 6 cm and 8 cm, multiply these by the ratio to get the leg lengths of the larger triangle: [ 6 cdot 2 = 12 text{ cm}, quad 8 cdot 2 = 16 text{ cm} ] 4. The perimeter of the larger triangle can be computed by adding the sides: [ 12 + 16 + 20 = boxed{48 text{ cm}} ]

answer:1. Given equation: [ |y+1| left(y^{2} + 2y + 28right) + |x-2| = 9 left(y^{2} + 2y + 4right) ] 2. Rewrite ( y^2 + 2y ) as ((y+1)^2 - 1). Let’s make this substitution: [ y^2 + 2y + 28 = (y+1)^2 - 1 + 28 = (y+1)^2 + 27 ] [ y^2 + 2y + 4 = (y+1)^2 - 1 + 4 = (y+1)^2 + 3 ] 3. Substituting these expressions into the given equation: [ |y+1| left((y+1)^2 + 27right) + |x-2| = 9 left((y+1)^2 + 3right) ] 4. Introduce new variables for simplicity: let ( x_1 = y + 1 ) and ( y_1 = x - 2 ). The equation becomes: [ |x_1| (x_1^2 + 27) + |y_1| = 9(x_1^2 + 3) ] 5. Next, solve for ( |y_1| ): [ |y_1| = 9(x_1^2 + 3) - |x_1| (x_1^2 + 27) ] 6. Factor out ((x_1^2 + 3)) from the expression on the right-hand side: [ |y_1| = (x_1^2 + 3) [9 - |x_1|] - 27 |x_1| ] 7. For the largest area of the rectangle with sides parallel to the coordinate axes, set up: [ x = 0 quad text{or} quad x = 3 quad text{(solving for other critical points too)} ] 8. Express the area function ( S(x) = -4x(x-3)^3 ): 9. Differentiate ( S(x) ) to find critical points: [ S'(x) = -4 left((x-3)^3 + 3x(x-3)^2 right) = -4(x-3)^2 (4x - 3) ] 10. Set the derivative equal to zero and solve for ( x ): [ -4 (x-3)^2 (4x-3) = 0 ] Critical points are ( x = 3 ) and ( x = frac{3}{4} ). 11. Analyze the critical points by substituting them back into ( S(x) ): [ S(0.75) = -4(0.75)(0.75 - 3)^3 = 34.171875 ] 12. Conclusion: [ boxed{34.171875} ]

answer:To solve this problem, we need to find the circumference of the original circular wire and then use that to find the radius. First, let's find the perimeter of the rectangle formed after cutting and bending the wire. Since the sides are in the ratio of 6:5, let's denote the longer side as 6x and the shorter side as 5x. We are given that the shorter side is 59.975859750350594 cm, so: 5x = 59.975859750350594 cm To find x, we divide the length of the shorter side by 5: x = 59.975859750350594 cm / 5 x = 11.995171950070119 cm Now we can find the length of the longer side: 6x = 6 * 11.995171950070119 cm 6x = 71.97103170042071 cm The perimeter (P) of the rectangle is the sum of all its sides: P = 2 * (length + width) P = 2 * (6x + 5x) P = 2 * (71.97103170042071 cm + 59.975859750350594 cm) P = 2 * (131.9468914507713 cm) P = 263.8937829015426 cm This perimeter is equal to the circumference of the original circle. The circumference (C) of a circle is given by the formula: C = 2 * π * r where r is the radius of the circle and π (pi) is approximately 3.14159. We can now set the perimeter equal to the circumference and solve for r: 263.8937829015426 cm = 2 * π * r Divide both sides by 2π to solve for r: r = 263.8937829015426 cm / (2 * π) r = 263.8937829015426 cm / (2 * 3.14159) r = 263.8937829015426 cm / 6.28318 r ≈ 42.00000000000001 cm So the radius of the original circular wire was approximately boxed{42} cm.

answer:To determine the number of different possible orders in which Harry, Ron, and Neville can finish the race given that there are no ties, we need to calculate the permutations of the three individuals. 1. **Identification of Permutation Problem:** We have three racers: Harry (H), Ron (R), and Neville (N). The problem requires the determination of how many different ways these three racers can finish the race without any ties. 2. **Mathematical Representation:** The scenario can be represented as finding the number of permutations of the set ({H, R, N}). 3. **Calculation of Permutations:** The number of permutations of (n) distinct objects is given by (n!). In this case, we have: [ 3! = 3 times 2 times 1 ] Calculating the factorial: [ 3! = 6 ] 4. **Verification by Listing All Possible Orders:** To verify, we can list all the possible permutations: [ begin{aligned} &1. , HNR &2. , HRN &3. , NHR &4. , NRH &5. , RHN &6. , RNH end{aligned} ] As listed, there are indeed 6 possible orders. # Conclusion: There are (6) different possible orders for Harry, Ron, and Neville to finish the race. [ boxed{6 text{ (B)}} ]