A building casts a shadow that is 40 feet long when the angle of elevation to the top of the building from the tip of the shadow is 30 degrees. How tall is the building?

To find the height of the building, we can use trigonometry. Specifically, we will use the tangent function, which relates the angle of elevation to the opposite side (the height of the building) and the adjacent side (the length of the shadow). ### Known Conditions: - Length of the shadow (adjacent side) = 40 feet - Angle of elevation = 30 degrees ### Step-by-Step Solution: 1. **Identify the relationship**: The tangent of the angle of elevation is given by the formula: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] where: - \(\theta\) is the angle of elevation (30 degrees), - "opposite" is the height of the building (let's denote it as \(h\)), - "adjacent" is the length of the shadow (40 feet). 2. **Set up the equation**: \[ \tan(30^\circ) = \frac{h}{40} \] 3. **Calculate \(\tan(30^\circ)\)**: The value of \(\tan(30^\circ)\) is \(\frac{1}{\sqrt{3}}\). 4. **Substitute and solve for \(h\)**: \[ \frac{1}{\sqrt{3}} = \frac{h}{40} \] To find \(h\), we can rearrange the equation: \[ h = 40 \cdot \frac{1}{\sqrt{3}} \] 5. **Calculate the height**: Now, we will calculate \(h\) using the expression derived. Let's perform the calculation. Calculate the value by following steps: - step0: Calculate: \(\frac{40}{\sqrt{3}}\) - step1: Multiply by the Conjugate: \(\frac{40\sqrt{3}}{\sqrt{3}\times \sqrt{3}}\) - step2: Multiply the numbers: \(\frac{40\sqrt{3}}{3}\) The height of the building can be calculated as follows: \[ h = \frac{40}{\sqrt{3}} \approx 23.094 \] ### Final Answer: The height of the building is approximately **23.1 feet**.