Two identical rectangular prisms each have a height of 90 centimeters \( (\mathrm{cm}) \). The base of each prism is a square, and the surface area of each prism is \( K \mathrm{~cm}^{2} \). If the prisms are glued together along a square base, the resulting prism has a surface area of \( \frac{92}{47} K \mathrm{~cm}^{2} \). What is the side length, in cm , of each square base?

Let's denote the side length of each square base as \( s \) cm. Given: - Height of each rectangular prism = 90 cm - Surface area of each rectangular prism = \( K \, \text{cm}^2 \) - Surface area of the resulting prism when the prisms are glued together = \( \frac{92}{47} K \, \text{cm}^2 \) The surface area of a rectangular prism is given by the formula: \[ \text{Surface Area} = 2lw + 2lh + 2wh \] where \( l \) is the length, \( w \) is the width, and \( h \) is the height. For each rectangular prism: - The base is a square, so \( l = w = s \) cm - The height is 90 cm Therefore, the surface area of each rectangular prism is: \[ K = 2s^2 + 2s(90) + 2s(90) \] \[ K = 2s^2 + 360s \] When the prisms are glued together along a square base, the resulting prism has a surface area of \( \frac{92}{47} K \, \text{cm}^2 \). The surface area of the resulting prism is: \[ \frac{92}{47} K = 2s^2 + 2s(90) + 2s(90) + 2s(90) + 2s(90) \] \[ \frac{92}{47} K = 2s^2 + 720s \] Now, we can set up an equation to solve for the side length \( s \) using the given information: \[ 2s^2 + 720s = \frac{92}{47} (2s^2 + 360s) \] Let's solve this equation to find the side length \( s \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(2s^{2}+720s=\frac{92}{47}\left(2s^{2}+360s\right)\) - step1: Multiply the terms: \(2s^{2}+720s=\frac{184}{47}s^{2}+\frac{33120}{47}s\) - step2: Move the expression to the left side: \(-\frac{90}{47}s^{2}+\frac{720}{47}s=0\) - step3: Factor the expression: \(-\frac{90}{47}s\left(s-8\right)=0\) - step4: Separate into possible cases: \(\begin{align}&s-8=0\\&-\frac{90}{47}s=0\end{align}\) - step5: Solve the equation: \(\begin{align}&s=8\\&s=0\end{align}\) - step6: Rewrite: \(s_{1}=0,s_{2}=8\) The solutions to the equation are \( s = 0 \) and \( s = 8 \). Since the side length of a square base cannot be 0, the side length of each square base is 8 cm.