A package with square ends has combined length and girth(girth is perimeter of a cross section) of 120 in. The surface area of the entire package is 3600 sq. in.
Determine the dimensions of the package.
s in. * s in.* l in.
One solution is: 11.08in. * 11.08in. * 75.68in. Find the other solution
Hint: if 4s+length=120 then length=120-4S
Round your answer to 2 decimals. Enter the 3 dimensions separated by commas
To find the dimensions of the package, we will solve the problem step by step.
Step 1: Let's assume that the side length of the square ends of the package is 's' inches.
Step 2: The girth of the package is the perimeter of a cross-section, which is equal to the sum of all four sides of the square. For a square, the girth is given by 4s.
Step 3: According to the problem, the combined length and girth of the package is 120 inches. So, we can express this as an equation: 4s + length = 120.
Step 4: We can also express the surface area of the package. The surface area of a rectangular prism is given by 2lw + 2lh + 2wh, where 'l' represents length, 'w' represents width, and 'h' represents height. In this case, we have two square ends with area s^2, and the rest of the surface area is given by length multiplied by the girth, which is length multiplied by 4s. So, the surface area equation is: 2s^2 + 4sl = 3600.
Step 5: Now we have two equations with two unknowns: 4s + length = 120 and 2s^2 + 4sl = 3600.
Step 6: Using the hint given in the problem, we know that length = 120 - 4s.
Step 7: Substitute this value of length in the surface area equation: 2s^2 + 4s(120 - 4s) = 3600.
Step 8: Simplify the equation: 2s^2 + 480s - 16s^2 = 3600.
-14s^2 + 480s - 3600 = 0.
Step 9: Solve this quadratic equation using any appropriate method. In this case, we can apply the quadratic formula.
Step 10: Apply the quadratic formula: s = (-b ± √(b^2 - 4ac)) / 2a.
s = (-480 ± √(480^2 - 4(-14)(-3600))) / 2(-14).
Step 11: Calculate the values: s ≈ 11.08 or s ≈ 25.92 (rounded to two decimal places).
Step 12: Since the dimensions of a package cannot be negative, we consider the positive value for 's', which is approximately 11.08 inches.
Step 13: Substitute this value of 's' in the equation length = 120 - 4s: length = 120 - 4(11.08) ≈ 75.68 inches.
Step 14: Therefore, the dimensions of the package are approximately 11.08 in. * 11.08 in. * 75.68 in.
And the other solution would be 25.92 in. * 25.92 in. * (120 - 4(25.92)) ≈ 25.92 in. * 25.92 in. * 24.32 in.
Rounding both solutions to 2 decimal places, we have: 11.08 in. * 11.08 in. * 75.68 in. and 25.92 in. * 25.92 in. * 24.32 in.