Type your answer If a car is climbing a hill at a \( 15^{\circ} \) angle, how many horizontal feet will the car travel every 1,000 feet of road if the car gains 259 feet of vertical feet? Round your answer to the nearest foot. (22 out 2 ?

To solve this problem, we can use trigonometry. We know that the car is climbing a hill at a \( 15^{\circ} \) angle and gains 259 feet of vertical height. We need to find out how many horizontal feet the car travels for every 1,000 feet of road. 1. **Identify the known values:** - Angle of incline, \( \theta = 15^{\circ} \) - Vertical gain, \( h = 259 \) feet 2. **Use the sine function to relate the vertical gain to the hypotenuse (the road traveled):** \[ \sin(\theta) = \frac{h}{d} \] where \( d \) is the distance traveled along the road. 3. **Rearranging the formula to find \( d \):** \[ d = \frac{h}{\sin(\theta)} \] 4. **Calculate \( d \) using the known values:** - First, we need to calculate \( \sin(15^{\circ}) \). 5. **Calculate the horizontal distance \( x \) using the cosine function:** \[ \cos(\theta) = \frac{x}{d} \] Rearranging gives: \[ x = d \cdot \cos(\theta) \] Now, let's perform the calculations step by step. First, we will calculate \( d \) and then \( x \). Let's calculate \( d \) and \( x \). Simplify the expression by following steps: - step0: Solution: \(\sin\left(15degre\times es\right)\) - step1: Rewrite the expression: \(\sin\left(15de^{3}grs\right)\) - step2: Multiply the numbers: \(\sin\left(15e^{3}dgrs\right)\) Calculate or simplify the expression \( \cos(15 degrees) \). Simplify the expression by following steps: - step0: Solution: \(\cos\left(15degre\times es\right)\) - step1: Rewrite the expression: \(\cos\left(15de^{3}grs\right)\) - step2: Multiply the numbers: \(\cos\left(15e^{3}dgrs\right)\) Calculate or simplify the expression \( \sin(15) \). Calculate the value by following steps: - step0: Calculate: \(\sin\left(15\right)\) Calculate or simplify the expression \( \cos(15) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(15\right)\) Calculate or simplify the expression \( 259/\cos(15) \). Calculate the value by following steps: - step0: Calculate: \(\frac{259}{\cos\left(15\right)}\) - step1: Calculate the trigonometric value: \(-340.929473\) Calculate or simplify the expression \( 259/\sin(15) \). Calculate the value by following steps: - step0: Calculate: \(\frac{259}{\sin\left(15\right)}\) - step1: Calculate the trigonometric value: \(398.285165\) Let's summarize the calculations step by step: 1. **Calculate \( \sin(15^{\circ}) \) and \( \cos(15^{\circ}) \):** - \( \sin(15^{\circ}) \approx 0.2588 \) - \( \cos(15^{\circ}) \approx 0.9659 \) 2. **Calculate the distance \( d \) (the road traveled) using the vertical gain:** \[ d = \frac{h}{\sin(15^{\circ})} = \frac{259}{0.2588} \approx 1000.77 \text{ feet} \] 3. **Calculate the horizontal distance \( x \):** \[ x = d \cdot \cos(15^{\circ}) = 1000.77 \cdot 0.9659 \approx 967.15 \text{ feet} \] 4. **Round \( x \) to the nearest foot:** \[ x \approx 967 \text{ feet} \] Thus, for every 1,000 feet of road, the car travels approximately **967 horizontal feet**.