Diagramming for Modeling - Math 2

1. A surveyor uses a clinometer to measure the angle of depression to a river from a cliff. Standing 180 feet above the river, she measures a 15° angle of depression to a point on the opposite bank. The horizontal distance from the base of her cliff to the river edge is 45 feet. Which labeling correctly identifies the setup for finding the width of the river?
   Answer: Known: height = 180 ft, angle of depression = 15°, cliff distance = 45 ft; Unknown: total distance = x, using tan(15°) = 180/x, then river width = x - 45 The angle of depression from the cliff to the opposite bank creates a right triangle. The height is 180 ft (opposite) and the total horizontal distance is x (adjacent). Using tan(15°) = 180/x gives the total distance, then subtract 45 ft to get river width.
2. A water tower's height is being measured using two observation points. From point A, 200 meters from the tower's base, the angle of elevation is 25°. From point B, 350 meters from the tower's base (on the opposite side), the angle of elevation is 18°. Which diagram labeling correctly identifies the relationship needed to verify the tower height using both measurements?
   Answer: Two separate right triangles: Triangle 1 has adjacent = 200m, angle = 25°; Triangle 2 has adjacent = 350m, angle = 18°; both should give same tower height using tangent Each observation point creates a separate right triangle with the tower. Point A: tan(25°) = h/200, Point B: tan(18°) = h/350. Both calculations should yield the same tower height h, providing verification of the measurement.
3. Two similar triangles are used to model the shadow cast by a flagpole. A 6-foot person casts a 4-foot shadow at the same time a flagpole casts a 28-foot shadow. Which proportion setup correctly identifies the known and unknown quantities for finding the flagpole height?
   Answer: $\frac{\text{person height}}{\text{person shadow}} = \frac{\text{flagpole height}}{\text{flagpole shadow}}$ where $\frac{6}{4} = \frac{h}{28}$ For similar triangles, corresponding sides are proportional. The ratio of height to shadow length must be the same for both objects. So person height/person shadow = flagpole height/flagpole shadow, giving 6/4 = h/28.
4. Two buildings of different heights cast shadows when the sun is at a 42° angle of elevation. Building A is 85 feet tall and casts a 90-foot shadow. Building B casts a 75-foot shadow. Which proportional setup correctly uses the known quantities to find Building B's height?
   Answer: $\frac{85}{90} = \frac{h}{75}$ where the ratio of height to shadow length is constant for both buildings Since both buildings are measured under the same sun angle, the ratio of height to shadow length must be the same for both buildings. Building A: 85/90, Building B: h/75. Setting them equal: 85/90 = h/75.

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