Question: Find the value of \( x \) so that the shaded figure to the right is a gnomon to the white…

Find the value of \( x \) so that the shaded figure to the right is a gnomon to the white rectangle. (Figure is not drawn to scale.)

\[ \begin{array}{c} \text{Figure:} \\ \begin{array}{ccc} \text{2.4} & & \\ \downarrow & & \downarrow \\ x & & x \\ \leftarrow & & \leftarrow \\ 16 & & 20 \\ \end{array} \end{array} \]

\( x = \boxed{\text{(Type an integer or a decimal.)}} \)

Solution

To find the value of \( x \) so that the shaded figure is a gnomon to the white rectangle, we need to understand that a gnomon is a figure that, when combined with the original figure, forms a new figure similar to the original. 1. The white rectangle has dimensions 16 (height) by 20 (width). 2. The entire shaded and white area together has dimensions \( 16 + 2x \) (height) by \( 20 + x \) (width). 3. To create a similar figure, the ratio of the sides of the larger rectangle must be equal to the ratio of the sides of the original rectangle. 4. Set up the proportion based on the dimensions: \[ \frac{16}{20} = \frac{16 + 2.4}{20 + x} \] 5. Solve the proportion for \( x \): \[ \frac{16}{20} = \frac{18.4}{20 + x} \] Cross-multiply: \[ 16(20 + x) = 20 \times 18.4 \] 6. Distribute and simplify: \[ 320 + 16x = 368 \] 7. Subtract 320 from both sides: \[ 16x = 48 \] 8. Divide both sides by 16: \[ x = 3 \] The value of \( x \) is 3.