|
|
| (9 intermediate revisions by 4 users not shown) |
| Line 1: |
Line 1: |
| − | == Problem ==
| + | #redirect[[2023 AMC 12B Problems/Problem 13]] |
| − | A rectangular box 𝒫 has distinct edge lengths 𝑎, 𝑏, and 𝑐. The sum of the lengths of
| |
| − | all 12 edges of 𝒫 is 13, the sum of the areas of all 6 faces of 𝒫 is <math>\dfrac{11}{2}</math>, and the volume of 𝒫 is <math>\dfrac{1}{2}</math>. What is the length of the longest interior diagonal connecting two vertices of 𝒫 ?
| |
| − | | |
| − | == Solution ==
| |
| − | | |
| − | <asy>
| |
| − | import geometry;
| |
| − | pair A = (-3, 4);
| |
| − | pair B = (-3, 5);
| |
| − | pair C = (-1, 4);
| |
| − | pair D = (-1, 5);
| |
| − | | |
| − | | |
| − | pair AA = (0, 0);
| |
| − | pair BB = (0, 1);
| |
| − | pair CC = (2, 0);
| |
| − | pair DD = (2, 1);
| |
| − | | |
| − | | |
| − | | |
| − | | |
| − | draw(D--AA,dashed);
| |
| − | | |
| − | draw(A--B);
| |
| − | draw(A--C);
| |
| − | draw(B--D);
| |
| − | draw(C--D);
| |
| − | | |
| − | draw(A--AA);
| |
| − | draw(B--BB);
| |
| − | draw(C--CC);
| |
| − | draw(D--DD);
| |
| − | | |
| − | // Dotted vertices
| |
| − | dot(A); dot(B); dot(C); dot(D);
| |
| − | | |
| − | | |
| − | | |
| − | dot(AA); dot(BB); dot(CC); dot(DD);
| |
| − | | |
| − | draw(AA--BB);
| |
| − | draw(AA--CC);
| |
| − | draw(BB--DD);
| |
| − | draw(CC--DD);
| |
| − | | |
| − | | |
| − | label("a",midpoint(D--DD),E);
| |
| − | label("b",midpoint(CC--DD),E);
| |
| − | label("c",midpoint(AA--CC),S);
| |
| − | </asy>
| |
| − | Let <math>a,b,</math> and <math>c</math> be the sides of the box, we get
| |
| − | | |
| − | <cmath>\begin{align*}
| |
| − | 4(a+b+c) &= 13\\
| |
| − | 2(ab+bc+ca) &= \dfrac{11}{2}\\
| |
| − | abc &= \dfrac{1}{2}
| |
| − | \end{align*}</cmath>
| |
| − | | |
| − | | |
| − | The diagonal of the box is
| |
| − | | |
| − | <cmath>\begin{align*}
| |
| − | \sqrt{a^2+b^2+c^2}&=\sqrt{(a+b+c)^2-2(ab+bc+ca)}\\
| |
| − | &=\sqrt{(\dfrac{13}{4})^2-\dfrac{11}{2}}\\
| |
| − | &=\sqrt{\dfrac{169}{16}-\dfrac{88}{16}}\\
| |
| − | &=\sqrt{\dfrac{81}{16}}\\
| |
| − | &=\dfrac{9}{4}
| |
| − | \end{align*}
| |
| − | </cmath>
| |
| − | | |
| − | ~Technodoggo
| |
