Let's break down the problem step by step: ### Given: - **Circumference of Earth:** \( C = 40,000,000 \) meters - **Extra rope added:** \( \Delta L = 10 \) meters - **Rope is now evenly raised above the Earth's surface** We are to find **how high above the Earth's surface the rope is**. --- ## Step 1: **Relationship Between Circumference and Radius** The circumference of a circle is given by: \[ C = 2\pi r \] where \( r \) is the radius. ### Initial Rope (Tight Around Earth): \[ C_1 = 2\pi r_1 = 40,000,000 \text{ meters} \] ### Rope with Extra 10 meters: \[ C_2 = 2\pi r_2 = 40,000,010 \text{ meters} \] --- ## Step 2: **Find the Change in Radius** Let \( h = r_2 - r_1 \) be the height above the surface. \[ C_2 - C_1 = 2\pi r_2 - 2\pi r_1 = 2\pi (r_2 - r_1) = 2\pi h \] \[ C_2 - C_1 = 10 \text{ meters} \] So, \[ 2\pi h = 10 \] \[ h = \frac{10}{2\pi} \] \[ h = \frac{10}{6.2832} \approx 1.59 \text{ meters} \] --- ## Step 3: **Interpret the Answer** The rope is approximately **1.59 meters** above the surface of the Earth. --- ## Step 4: **Select the Right Option** - A child would be able to walk under the rope without bending down. (**TRUE**) - A hydrogen molecule would fit under the rope, but only sideways. (Not relevant, as 1.59 meters is much larger than a molecule) - A cat would be able to walk under the rope. (TRUE, but a child is a better answer) - An adult giraffe would be able to walk under the rope. (Giraffes are ~5-6 meters tall, so FALSE) - A small ant would be able to crawl under the rope. (TRUE, but again the child answer is better) ### **Best Answer:** **A child would be able to walk under the rope without bending down.** --- ## **Final Answer:** > **A child would be able to walk under the rope without bending down.** ### **Key Calculation:** \[ h = \frac{10}{2\pi} \approx 1.59 \text{ meters} \]

Let's go through the problem step by step with detailed calculations and explanations for each option, leading to the final answer. --- ### **Problem Restatement:** - The Earth's circumference at the equator is approximately **40 million meters**. - A rope is tied around the Earth at the equator, just tight enough to fit (no slack). - Then, **10 meters** of extra rope are added evenly around the entire circumference. - This extra length causes the rope to rise above the Earth's surface. - Minions are then used to lift the rope evenly, raising it uniformly by some height \( h \). **Question:** How high is the rope above the Earth's surface after adding the extra length? --- ### **Step 1: Understand the initial data** - Initial circumference: \[ C_1 = 40,000,000 \text{ meters} \] - Extra rope added: \[ \Delta L = 10 \text{ meters} \] - New circumference: \[ C_2 = C_1 + \Delta L = 40,000,010 \text{ meters} \] --- ### **Step 2: Relate circumference to radius** The circumference of a circle: \[ C = 2\pi r \] - Initial radius: \[ r_1 = \frac{C_1}{2\pi} \] - Final radius after adding the extra length: \[ r_2 = \frac{C_2}{2\pi} \] The increase in radius (which is the height \( h \) the rope is lifted): \[ h = r_2 - r_1 = \frac{C_2 - C_1}{2\pi} \] --- ### **Step 3: Calculate the height \( h \)** \[ h = \frac{10}{2\pi} \] Calculating: \[ 2\pi \approx 6.2832 \] \[ h = \frac{10}{6.2832} \approx 1.59 \text{ meters} \] **Therefore, the rope rises approximately 1.59 meters above the Earth's surface.** --- ### **Step 4: Interpretation of options** Now, let's analyze each option based on the height: 1. **A child would be able to walk under the rope without bending down.** - **Child's height:** approximately 1.2 meters. - Since \( h \approx 1.59 \) meters, a child can walk under without bending. 2. **A hydrogen molecule would fit under the rope, but only sideways.** - The size of a hydrogen molecule is on the order of **0.1 nanometers** (~\(10^{-10}\) meters), which is negligible. - The statement is technically true but trivial; the molecule is far too small to be relevant here. 3. **A cat would be able to walk under the rope.** - **Cat height:** approximately 0.3 meters. - Since \( 1.59 \text{ meters} \gg 0.3 \text{ meters} \), a cat can easily walk under. 4. **An adult giraffe would be able to walk under the rope.** - **Giraffe height:** approximately 5-6 meters. - Since \( 1.59 \text{ meters} < 5 \text{ meters} \), a giraffe **cannot** walk under without bending down. 5. **A small ant would be able to crawl under the rope.** - **Ant's height:** approximately 0.01 meters or smaller. - Definitely true; an ant can crawl under easily. --- ### **Final conclusion:** - The **most accurate and relevant** answer, considering the height, is: - **A child would be able to walk under the rope without bending down.** --- ### **### Final Answer:** **A child would be able to walk under the rope without bending down.** --- ### **Summary of key calculations:** \[ h = \frac{\text{Extra length}}{2 \pi} = \frac{10}{6.2832} \approx 1.59 \text{ meters} \] which confirms the height above surface, and helps us evaluate the options correctly. --- **If you have further questions or need more clarification, feel free to ask!**