Question

Hey can anyone help me with this

Answer 1

Answer:

$5\sqrt{3}$, or 8.7

Step-by-step explanation:

To solve this problem, you need to use the Pythagorean theorem $a^{2} +b^{2} =c^{2}$. The tricky part is visualizing where the right triangle will be in the cube. The diagonal you are trying to find will be the hypotenuse, the diagonal of the base of the cube is one leg, and the edge of the cube is another leg.

First, you need to find the lengths of both legs of the triangle. The first leg is the length of the edge of the cube, which is given as 5 cm.

The next leg is the diagonal of the base of the cube, which is a 5x5 square. To find the diagonal of the 5x5 square, you can cut the square across its diagonal and use the 45-45-90 special right triangle rule. Each leg of the newly formed 45-45-90 triangle is 5 cm, which can be represented by x, and the hypotenuse, which is also the diagonal of the original 5x5 square, can be represented as $x\sqrt{2}$.

Using 5 as x and substituting, you can find that the diagonal of the 5x5 square is $5\sqrt{2}$ cm, so the second leg of the right triangle is $5\sqrt{2}$ cm.

Now, you can do a final substitution of both legs into the Pythagorean theorem, which turns out as $(5\sqrt{2})^{2} +5^{2} =c^{2}$

Solving this equation starts with doing out the squares on the left side as $50+25=c^{2}$ which means $c^{2} =75$. From here you take the square root of both sides of the equation to get $c=\sqrt{75}$ or $c=5\sqrt{3}$.

The final step is to make this solution rational by rounding it, so plug in $5\sqrt{3}$ into a calculator and round to the nearest tenth to get 8.7.