Question Anand is 1.75 meters tall. At $$ 10 \mathrm{a} . \mathrm{m} $$., he measures the length of a tree's shadow to be 41.35 meters. He stands 36.9 meters away from the tree, so that the tip of his shadow meets the tip of the tree's shadow. Find the height of the tree to the nearest hundredth of a meter.

Question

UpStudy AI · Accepted Answer

To find the height of the tree, we can use the concept of similar triangles. The triangles formed by Anand and the tree with their respective shadows are similar because they are both right triangles sharing the same angle of elevation from the ground to the top of the objects.

### Known Conditions:
- Height of Anand ($$ h_a $$): 1.75 meters
- Length of Anand's shadow ($$ s_a $$): 41.35 meters
- Distance from Anand to the tree ($$ d $$): 36.9 meters
- Length of the tree's shadow ($$ s_t $$): $$ s_a + d $$

### Step-by-Step Solution:

1. **Calculate the total length of the tree's shadow**:
   $$
   s_t = s_a + d = 41.35 + 36.9
   $$

2. **Set up the proportion using similar triangles**:
   $$
   \frac{h_a}{s_a} = \frac{h_t}{s_t}
   $$
   where $$ h_t $$ is the height of the tree.

3. **Rearranging the proportion to solve for $$ h_t $$**:
   $$
   h_t = h_a \cdot \frac{s_t}{s_a}
   $$

Now, let's calculate the values step by step.

### Calculating the total length of the tree's shadow:
$$
s_t = 41.35 + 36.9
$$

### Now, let's calculate $$ h_t $$:
$$
h_t = 1.75 \cdot \frac{s_t}{41.35}
$$

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$41.35+36.9$$
- step1: Add the numbers:
  $$78.25$$
Calculate or simplify the expression $$ 1.75 * (s_t / 41.35) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$1.75\left(\frac{s_{t}}{41.35}\right)$$
- step1: Divide the terms:
  $$1.75\times \frac{20s_{t}}{827}$$
- step2: Rewrite the expression:
  $$\frac{7}{4}\times \frac{20s_{t}}{827}$$
- step3: Reduce the fraction:
  $$7\times \frac{5s_{t}}{827}$$
- step4: Multiply the terms:
  $$\frac{7\times 5s_{t}}{827}$$
- step5: Multiply the terms:
  $$\frac{35s_{t}}{827}$$
Calculate or simplify the expression $$ 1.75 * (78.25 / 41.35) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1.75\left(\frac{78.25}{41.35}\right)$$
- step1: Divide the terms:
  $$1.75\times \frac{1565}{827}$$
- step2: Convert the expressions:
  $$\frac{7}{4}\times \frac{1565}{827}$$
- step3: Multiply the numbers:
  $$\frac{10955}{3308}$$
### Step 1: Calculate the total length of the tree's shadow
The total length of the tree's shadow is:
$$
s_t = 41.35 + 36.9 = 78.25 \text{ meters}
$$

### Step 2: Set up the proportion and calculate the height of the tree
Using the proportion of similar triangles:
$$
h_t = 1.75 \cdot \frac{s_t}{s_a} = 1.75 \cdot \frac{78.25}{41.35}
$$

Calculating $$ h_t $$:
$$
h_t \approx 1.75 \cdot 1.895 = 3.311669 \text{ meters}
$$

### Final Result
Rounding to the nearest hundredth of a meter, the height of the tree is:
$$
\boxed{3.31} \text{ meters}
$$
The height of the tree is approximately 3.31 meters.

Question
Anand is 1.75 meters tall. At ., he measures the length of a tree’s shadow to be 41.35 meters. He stands 36.9 meters
away from the tree, so that the tip of his shadow meets the tip of the tree’s shadow. Find the height of the tree to the nearest
hundredth of a meter.

Upstudy AI Solution

Answer

Solution

Known Conditions:

Step-by-Step Solution:

Calculating the total length of the tree’s shadow:

Now, let’s calculate :

Step 1: Calculate the total length of the tree’s shadow

Step 2: Set up the proportion and calculate the height of the tree

Final Result

Beyond the Answer

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Question Anand is 1.75 meters tall. At ., he measures the length of a tree’s shadow to be 41.35 meters. He stands 36.9 meters away from the tree, so that the tip of his shadow meets the tip of the tree’s shadow. Find the height of the tree to the nearest hundredth of a meter.