2) The angle of elevation to the top of a building is \( 42^{\circ} \). If the observer moves of elevation is \( 65^{\circ} \). How tall is the building? Round to two decimal places. State the smallest positive and greatest negative angle co-terminal with \( \theta=-1535^{\circ} \)

To solve the problem, we will break it down into two parts: finding the height of the building and determining the co-terminal angles. ### Part 1: Finding the Height of the Building Let: - \( h \) = height of the building - \( d \) = horizontal distance from the observer to the base of the building From the first position, the angle of elevation to the top of the building is \( 42^\circ \). Using the tangent function, we have: \[ \tan(42^\circ) = \frac{h}{d} \] From this, we can express \( h \) in terms of \( d \): \[ h = d \cdot \tan(42^\circ) \] After moving a certain distance away from the building, the angle of elevation becomes \( 65^\circ \). Let’s denote the new horizontal distance as \( d' \). The new distance \( d' \) can be expressed as: \[ d' = d + x \] where \( x \) is the distance moved away from the building. The relationship for the new angle of elevation is: \[ \tan(65^\circ) = \frac{h}{d'} \] Substituting \( d' \): \[ \tan(65^\circ) = \frac{h}{d + x} \] Now we have two equations: 1. \( h = d \cdot \tan(42^\circ) \) 2. \( h = (d + x) \cdot \tan(65^\circ) \) Setting these equal to each other gives: \[ d \cdot \tan(42^\circ) = (d + x) \cdot \tan(65^\circ) \] Now, we can solve for \( d \): \[ d \cdot \tan(42^\circ) = d \cdot \tan(65^\circ) + x \cdot \tan(65^\circ) \] Rearranging gives: \[ d \cdot (\tan(42^\circ) - \tan(65^\circ)) = x \cdot \tan(65^\circ) \] Thus, \[ d = \frac{x \cdot \tan(65^\circ)}{\tan(42^\circ) - \tan(65^\circ)} \] Now substituting \( d \) back into the equation for \( h \): \[ h = d \cdot \tan(42^\circ) = \frac{x \cdot \tan(65^\circ)}{\tan(42^\circ) - \tan(65^\circ)} \cdot \tan(42^\circ) \] Calculating the values of the tangents: - \( \tan(42^\circ) \approx 0.9004 \) - \( \tan(65^\circ) \approx 2.1445 \) Substituting these values: \[ h = \frac{x \cdot 2.1445}{0.9004 - 2.1445} \cdot 0.9004 \] Calculating the denominator: \[ 0.9004 - 2.1445 \approx -1.2441 \] Thus, \[ h = \frac{x \cdot 2.1445 \cdot 0.9004}{-1.2441} \] This gives us the height \( h \) in terms of \( x \). To find a specific height, we need the value of \( x \). If \( x \) is provided, we can calculate \( h \). ### Part 2: Co-terminal Angles To find the smallest positive and greatest negative angle co-terminal with \( \theta = -1535^\circ \), we can use the formula for co-terminal angles: \[ \theta + 360k \] where \( k \) is an integer. 1. **Smallest Positive Angle**: We can find the smallest positive angle by adding \( 360^\circ \) until we get a positive angle: \[ -1535 + 360 \cdot 5 = -1535 + 1800 = 265^\circ \] So, the smallest positive co-terminal angle is \( 265^\circ \). 2. **Greatest Negative Angle**: We can find the greatest negative angle by adding \( 360^\circ \) until we get a negative angle: \[ -1535 + 360 \cdot 4 = -1535 + 1440 = -95^\circ \] So, the greatest negative co-terminal angle is \( -95^\circ \). ### Summary - The height of the building \( h \) can be expressed in terms of \( x \) as: \[ h = \frac{x \cdot 2.1445 \cdot 0.9004}{-1.2441} \] - The smallest positive co-terminal angle is \( 265^\circ \). - The greatest negative co-terminal angle is \( -95^\circ \).