Armena casts a 7-foot shadow at the same time a nearby mailbox casts a 4-foot shadow. If the mailbox is 3 feet tall, how tall is Armena?

Answer 1

To find the height of Armena, we can use the concept of similar triangles, as the lengths of the shadows and the heights of the objects are proportional.

Let \( h \) be the height of Armena. The ratio of Armena's height to the length of her shadow should equal the ratio of the mailbox's height to the length of its shadow.

We can set up the proportion as follows:

\[ \frac{h}{7} = \frac{3}{4} \]

Now, we can cross-multiply to solve for \( h \):

\[ 4h = 3 \times 7 \]

Calculating the right side:

\[ 4h = 21 \]

Now, divide both sides by 4 to isolate \( h \):

\[ h = \frac{21}{4} \]

Calculating that gives:

\[ h = 5.25 \]

Thus, Armena's height is \( 5.25 \) feet.

Answer 2

Water ride, Roller Coaster, and Ferris Wheel are at vertices of triangle on left. Park Entrance, Information Booth, Ferris Wheel at vertices of triangle on right. Water ride to roller coaster = d meters. Water ride to Ferris wheel = 45 m. Park entrance to information booth = 10 m. Park Entrance to Ferris Wheel = 21 m.

In the figure, the triangles are similar. What is the distance from the water ride to the roller coaster? Round to the nearest tenth.

Answer 3

Since the triangles formed by the attractions are similar, we can set up a proportion based on their corresponding sides.

Let's denote:

• \( d \): the distance from the Water Ride to the Roller Coaster (this is what we're trying to find).
• The distance from the Water Ride to the Ferris Wheel is given as \( 45 \) m.
• The distance from the Park Entrance to the Information Booth is given as \( 10 \) m.
• The distance from the Park Entrance to the Ferris Wheel is given as \( 21 \) m.

Since the triangles are similar, we can set up the following proportion:

\[ \frac{d}{45} = \frac{10}{21} \]

Now, we can cross-multiply to solve for \( d \):

\[ 21d = 10 \times 45 \]

Calculating the right side:

\[ 21d = 450 \]

Now, divide both sides by \( 21 \):

\[ d = \frac{450}{21} \approx 21.4286 \]

When rounded to the nearest tenth, we have:

\[ d \approx 21.4 \text{ meters} \]

Thus, the distance from the Water Ride to the Roller Coaster is approximately 21.4 meters.