Jessica is making an entry table in woodshop. She designs the table so the back piece has an X formation between the two back legs. The height of the legs is 2.5 feet and the diagonal distance from the top of one leg to the bottom of the other is 5 feet. Using the inverse of cosine what is the approximate angle formed where the X formation meets the top of the table? Round your answer to the nearest whole degree

1 answer

To find the angle formed where the X formation meets the top of the table, we can use some trigonometric relationships. Specifically, we can use the cosine function.

Let's denote:

  • \( h = 2.5 \) feet (the height of the legs),
  • \( d = 5 \) feet (the diagonal distance from the top of one leg to the bottom of the other).

We can visualize this setup as a right triangle where:

  • One leg is the height \( h \),
  • The other leg is the horizontal distance \( x \) between the two legs at the table's top.

Using the Pythagorean theorem, we can relate the diagonal distance \( d \), height \( h \), and horizontal distance \( x \):

\[ d^2 = h^2 + x^2 \]

Substituting the values we have:

\[ 5^2 = (2.5)^2 + x^2 \] \[ 25 = 6.25 + x^2 \] \[ x^2 = 25 - 6.25 \] \[ x^2 = 18.75 \] \[ x = \sqrt{18.75} \] \[ x \approx 4.33 \text{ feet} \]

Now, we can find the angle \( \theta \) at which the X formation meets the top of the table using the cosine function:

\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{h}{d} \]

Substituting the known values:

\[ \cos(\theta) = \frac{2.5}{5} = 0.5 \]

Now, we take the inverse cosine to find the angle:

\[ \theta = \cos^{-1}(0.5) \]

The cosine inverse of 0.5 corresponds to:

\[ \theta = 60^\circ \]

Therefore, rounding to the nearest whole degree, the approximate angle formed where the X formation meets the top of the table is:

\[ \boxed{60} \] degrees.