Question

a giraffe is looking down at a bag of peanuts while a boy, sitting on his dad’s shoulders, is looking up at the giraffe. the distance from the ground to the giraffe’s eye is 12 feet, and the giraffe is 48 feet away from the bag of peanuts. the diagram is illustrated using similar triangles. the boy and his dad are 28 feet away from the bag of peanuts.
note: image not drawn to scale
complete the sentence.
the height of the boy and his dad is □ feet

Explanation:

Step1: Identify similar triangles

The problem involves two similar right triangles. Let \( h \) be the height of the boy and his dad. The larger triangle has height \( 12 + h \) (giraffe's eye height plus their height) and base 48 ft. The smaller triangle has height 12 ft and base \( 48 - 28 = 20 \) ft? Wait, no, recheck: The distance from the bag to the giraffe is 48 ft, and from the bag to the boy/dad is 28 ft? Wait, no, the boy and dad are 28 ft away from the bag? Wait, the diagram: the larger triangle's base is 48 ft (distance from giraffe to bag), the smaller triangle's base is \( 48 - 28 = 20 \) ft? No, maybe the two triangles are similar, so the ratios of corresponding sides are equal. Let's define:

Let \( h \) = height of boy and dad.

The giraffe's eye height is 12 ft above some reference? Wait, the larger triangle: height is \( 12 + h \)? No, wait, the vertical side: the giraffe's eye is at height \( h + 12 \)? Wait, no, the boy and dad's height is \( h \), and the giraffe's eye is 12 ft above their height? Wait, the diagram shows a right triangle with height 12 ft (giraffe's eye above the boy/dad's height) and base 48 ft (distance from giraffe to bag), and another right triangle with height \( h \) (boy/dad's height) and base 28 ft (distance from boy/dad to bag)? Wait, no, similar triangles: corresponding angles are equal, so the ratio of height to base should be equal.

Wait, correct setup: The two triangles are similar, so \( \frac{h}{28} = \frac{12 + h}{48} \)? No, that can't be. Wait, maybe the smaller triangle is height 12 ft, base \( 48 - 28 = 20 \) ft? Wait, no, let's read the problem again:

"A giraffe is looking down at a bag of peanuts while a boy, sitting on his dad’s shoulders, is looking up at the giraffe. The distance from the ground to the giraffe’s eye is 12 feet, and the bag of peanuts is 48 feet away from the giraffe. The boy and his dad are 28 feet away from the bag of peanuts. The diagram is illustrated using similar triangles."

So, the two similar triangles:

1. Larger triangle: height = 12 ft (giraffe's eye height) + \( h \) (boy/dad height)? No, wait, the giraffe's eye is at height 12 ft above the ground? No, the boy and dad are on the ground? Wait, no, the boy is on his dad's shoulders, so their height is \( h \), and the giraffe's eye is 12 ft above their height? Wait, the vertical distance from their height to the giraffe's eye is 12 ft. The horizontal distance from the giraffe to the bag is 48 ft, and from the boy/dad to the bag is 28 ft, so the horizontal distance from the giraffe to the boy/dad is \( 48 - 28 = 20 \) ft.

So, the two similar right triangles:

- Triangle 1: vertical leg = 12 ft, horizontal leg = 20 ft (distance from giraffe to boy/dad: 48 - 28 = 20 ft)
- Triangle 2: vertical leg = \( h \) (boy/dad height), horizontal leg = 28 ft (distance from boy/dad to bag)

Wait, no, similar triangles: the angles are equal, so the ratio of vertical to horizontal should be equal. So \( \frac{12}{20} = \frac{h}{28} \)

Wait, that makes sense: the giraffe's line of sight forms a triangle with vertical 12 and horizontal 20 (48 - 28), and the boy's line of sight forms a triangle with vertical \( h \) and horizontal 28.

So, solving \( \frac{12}{20} = \frac{h}{28} \)

Step2: Solve for \( h \)

Cross-multiplying: \( 20h = 12 \times 28 \)

Calculate \( 12 \times 28 = 336 \)

Then \( h = \frac{336}{20} = 16.8 \)? Wait, no, that can't be. Wait, maybe I mixed up the triangles.

Wait, another approach: The two triangles are similar, so the ratio of height to base is equal. The larger triangle (giraffe to bag) has height \( h + 12 \) (boy/dad height + 12 ft) and base 48 ft. The smaller triangle (boy/dad to bag) has height \( h \) and base 28 ft. Wait, no, that would be \( \frac{h + 12}{48} = \frac{h}{28} \)

Cross-multiplying: \( 28(h + 12) = 48h \)

\( 28h + 336 = 48h \)

\( 336 = 20h \)

\( h = \frac{336}{20} = 16.8 \). Wait, but that seems low. Wait, maybe the giraffe's eye height is 12 ft above the boy/dad's height, so the vertical leg is 12, horizontal leg is 48 - 28 = 20, and the boy/dad's height is \( h \), horizontal leg 28. So similar triangles: \( \frac{12}{20} = \frac{h}{28} \)

Yes, that's correct. So \( h = \frac{12 \times 28}{20} = \frac{336}{20} = 16.8 \)? Wait, but maybe the diagram is different. Wait, the problem says "the height of the boy and his dad is" and we need to find it. Wait, maybe I made a mistake in the base. Let's re-express:

Distance from giraffe to bag: 48 ft.

Distance from boy/dad to bag: 28 ft.

So distance from giraffe to boy/dad: 48 - 28 = 20 ft.

The two similar triangles:

- Giraffe's triangle: height = 12 ft (vertical), base = 20 ft (horizontal)
- Boy/dad's triangle: height = \( h \) (vertical), base = 28 ft (horizontal)

Since they are similar, \( \frac{12}{20} = \frac{h}{28} \)

Solving for \( h \):

\( h = \frac{12 \times 28}{20} = \frac{336}{20} = 16.8 \) feet? Wait, but 16.8 is 16.8, but maybe the problem expects integer? Wait, maybe I flipped the ratio.

Wait, maybe the larger triangle is the one with base 48 and height \( h + 12 \), and the smaller with base 28 and height \( h \). Then \( \frac{h + 12}{48} = \frac{h}{28} \)

Cross-multiplying: 28(h + 12) = 48h

28h + 336 = 48h

336 = 20h

h = 16.8. So that's the height.

Answer:

\( 16.8 \) (or if we consider fractions, \( \frac{84}{5} \) or 16.8)