Final answer:
The value of n, where two cuboids, one with dimensions (1, n, 15) and the other (n, n, 7), have the same surface area, is found by setting up and solving a quadratic equation. The solution is n = 6.
Step-by-step explanation:
Two cuboids have the same surface area; one is sized (1, n, 15) and the other is (n, n, 7). To determine the value of n, we start by writing the equation for the surface area (SA) of each cuboid.
For the first cuboid, SA = 2(n*1) + 2(n*15) + 2(1*15) = 2n + 30n + 30 = 32n + 30.
For the second cuboid, SA = 2(n*n) + 2(n*7) + 2(n*7) = 2n² + 28n.
Since their surface areas are equal, the following equation holds: 32n + 30 = 2n² + 28n.
Solving for n, we get: 2n² - 4n - 30 = 0. Factoring the quadratic equation, we can find the value of n.
2n² - 4n - 30 can be factorized as (2n + 5)(n - 6) = 0, which gives us n = -5/2 (which we discard, since a cuboid cannot have a negative dimensional value) or n = 6. Hence, n = 6 is the solution we seek.