Nathan is standing next to a building. He is 6 feet tall and the length of his shadow is 9 feet. If the length of the shadow of the building is 322.5 feet, how tall is the building?

Answer 1

To find the height of the building given the lengths of the shadows, we can use similar triangles. The proportion can be set up based on the heights and shadow lengths.

Let \( h \) be the height of the building. We know:

• Nathan's height: 6 feet
• Nathan's shadow: 9 feet
• Building's shadow: 322.5 feet

We can set up the proportion:

\[ \frac{h}{322.5} = \frac{6}{9} \]

Now, we simplify the fraction on the right:

\[ \frac{6}{9} = \frac{2}{3} \]

So we now have:

\[ \frac{h}{322.5} = \frac{2}{3} \]

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

\[ 3h = 2 \times 322.5 \]

Calculating \( 2 \times 322.5 \):

\[ 2 \times 322.5 = 645 \]

Now substitute back into the equation:

\[ 3h = 645 \]

Dividing both sides by 3 to isolate \( h \):

\[ h = \frac{645}{3} = 215 \]

Therefore, the height of the building is \( \boxed{215} \) feet.

Answer 2

A cell phone tower casts a 100-foot shadow. At the same time, a 4-foot 6-inch post near the tower casts a shadow of 3 feet 4 inches. How tall is the cell phone tower?

Answer 3

To find the height of the cell phone tower, we can use the concept of similar triangles, just like in the previous example.

First, let's convert all measurements to the same unit. We'll convert everything to feet.

• The height of the post is 4 feet 6 inches. In feet, this converts to: \[ 4 \text{ feet} + \frac{6 \text{ inches}}{12} = 4 + 0.5 = 4.5 \text{ feet} \]

• The length of the post's shadow is 3 feet 4 inches. In feet, this converts to: \[ 3 \text{ feet} + \frac{4 \text{ inches}}{12} = 3 + \frac{1}{3} = 3.333 \text{ feet} \]

Now, we can set up a proportion based on the height and shadow length:

Let \( h \) be the height of the cell phone tower. We know:

• The height of the post: 4.5 feet
• The shadow of the post: 3.333 feet
• The shadow of the tower: 100 feet

The proportion is as follows:

\[ \frac{h}{100} = \frac{4.5}{3.333} \]

Now, we will solve for \( h \). First, we can simplify the right side:

\[ \frac{4.5}{3.333} \approx 1.3515 \]

Cross-multiplying gives us:

\[ h = 100 \times \frac{4.5}{3.333} \]

Calculating \( h \):

\[ h \approx 100 \times 1.3515 \approx 135.15 \]

To be more precise, let's calculate \( \frac{4.5}{3.333} \) exactly:

Using the fraction: \[ 4.5 \div 3.333 \approx 1.35135 \]

Now using that in our equation:

\[ h \approx 100 \times 1.35135 \approx 135.135 \]

Since we can round this to two decimal places, we get:

\[ h \approx 135.14 \text{ feet} \]

Thus, the height of the cell phone tower is approximately \( \boxed{135.14} \) feet.

Answer 4

no cross multiply

Answer 5

Understood! Let's solve the problem again without using cross multiplication.

We are given:

• The height of the post is 4 feet 6 inches, which is 4.5 feet.
• The shadow of the post is 3 feet 4 inches, which is approximately 3.33 feet (or more precisely, \( \frac{10}{3} \) feet).
• The shadow of the cell phone tower is 100 feet.

We can set up the ratio of the height of the post to the length of its shadow and the height of the tower to the length of its shadow:

\[ \frac{\text{Height of Post}}{\text{Shadow of Post}} = \frac{\text{Height of Tower}}{\text{Shadow of Tower}} \]

Substituting the values we have:

\[ \frac{4.5}{3.33} = \frac{h}{100} \]

To find \( h \) (the height of the tower), we can express the equation as:

\[ h = \frac{4.5}{3.33} \times 100 \]

Next, let's calculate \( \frac{4.5}{3.33} \):

First, we approximate:

\[ \frac{4.5}{3.33} \approx 1.3514 \]

Now, we can calculate \( h \):

\[ h \approx 1.3514 \times 100 \approx 135.14 \text{ feet} \]

Thus, the height of the cell phone tower is approximately \( \boxed{135.14} \) feet.

To avoid approximation errors, let's simplify \( \frac{4.5}{\frac{10}{3}} \):

\[ \frac{4.5 \times 3}{10} = \frac{13.5}{10} = 1.35 \] Now use:

\[ h = 1.35 \times 100 = 135 \text{ feet} \]

So the height of the cell phone tower is \( \boxed{135} \) feet.

Answer 6

A cell phone tower casts a 100-foot shadow. At the same time, a 4-foot 6-inch post near the tower casts a shadow of 3 feet 4 inches. How tall is the cell phone tower?

Answer 7

To find the height of the cell phone tower using the information given, we can set up a proportion based on the similar triangles formed by the objects and their shadows.

Given:

• Height of the post: 4 feet 6 inches
• Shadow of the post: 3 feet 4 inches
• Shadow of the tower: 100 feet

Convert Measurements:

1. Post's Height: \[ 4 \text{ feet} + 6 \text{ inches} = 4 + \frac{6}{12} = 4 + 0.5 = 4.5 \text{ feet} \]

2. Post's Shadow: \[ 3 \text{ feet} + 4 \text{ inches} = 3 + \frac{4}{12} = 3 + \frac{1}{3} = 3.33 \text{ feet} \quad (\text{or } \frac{10}{3} \text{ feet}) \]

Now, we set up the proportion using the heights and lengths of the shadows:

\[ \frac{\text{Height of Post}}{\text{Shadow of Post}} = \frac{\text{Height of Tower}}{\text{Shadow of Tower}} \]

Substituting the values:

\[ \frac{4.5}{3.33} = \frac{h}{100} \]

Solve for \( h \):

1. To find \( h \), we can rearrange the formula, multiplying both sides by 100:

\[ h = 100 \times \frac{4.5}{3.33} \]

2. Calculate \( \frac{4.5}{3.33} \):

Using the approximate value \( 3.33 \approx \frac{10}{3} \), we can calculate:

\[ \frac{4.5}{3.33} \approx \frac{4.5 \times 3}{10} = \frac{13.5}{10} = 1.35 \]

3. Substitute back to find \( h \):

\[ h = 100 \times 1.35 = 135 \]

Thus, the height of the cell phone tower is approximately \( \boxed{135} \) feet.

Answer 8

When using shadows to determine height, why is it important that the lengths of the shadows are measured at the same time?

Answer 9

When using shadows to determine heights, it is crucial that the lengths of the shadows are measured at the same time for several reasons:

1. Consistency in Sun Position: The angle and position of the sun in the sky change throughout the day. If shadows are cast at different times, the angle of sunlight changes, which results in different lengths of shadows for the same height object. Measuring the lengths of shadows at the same time ensures that the position of the sun is identical, allowing for accurate comparisons.

2. Proportional Relationships: The use of similar triangles depends on the proportionality of the objects' heights to their respective shadow lengths. Consistent conditions (same time of day) maintain that this proportionality holds true. If shadows are measured at different times, the ratio would not accurately reflect the height relationship.

3. Environmental Factors: Factors like obstacles, terrain, and weather can also affect the length of shadows. For example, clouds might diffuse sunlight and slightly change shadow lengths, or an object might block direct light. Measuring at the same time minimizes these external influences and gives a clearer relationship between heights and shadows.

4. Accuracy and Precision: Consistency in measurements improves the accuracy and reliability of the results. When measuring at the same time, you're less likely to encounter random variables that could introduce errors.

Overall, measuring the lengths of the shadows at the same time ensures that the data used for calculations maintains the necessary conditions for valid results, allowing for reliable height estimations based on shadowing techniques.