7. A scoop of ice cream has a 3 -inch diameter. How tall should the ice cream cone of the same diameter be in order to contain all of the ice cream inside the cone?

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To determine how tall the ice cream cone should be to contain all of the ice cream, we'll follow these steps:

1. **Understand the Problem:**
   - **Ice Cream Scoop:** Assumed to be a sphere with a diameter of **3 inches**.
   - **Cone:** Has the same diameter (3 inches) as the scoop and needs to hold the entire volume of the scoop.

2. **Calculate the Volume of the Ice Cream Scoop (Sphere):**
   - **Radius of Sphere (rₛ):** Half of the diameter = 3 inches / 2 = **1.5 inches**.
   - **Volume of Sphere (Vₛ):**
     $$
     Vₛ = \frac{4}{3} \pi rₛ^3 = \frac{4}{3} \pi (1.5)^3 = \frac{4}{3} \pi (3.375) = 4.5\pi \text{ cubic inches}
     $$

3. **Set Up the Volume of the Cone:**
   - **Radius of Cone (r_c):** Same as the sphere = **1.5 inches**.
   - **Height of Cone (h):** What we need to find.
   - **Volume of Cone (V_c):**
     $$
     V_c = \frac{1}{3} \pi r_c^2 h = \frac{1}{3} \pi (1.5)^2 h = \frac{1}{3} \pi (2.25) h = 0.75\pi h \text{ cubic inches}
     $$

4. **Equate the Volumes to Solve for Height (h):**
   $$
   V_c = Vₛ \Rightarrow 0.75\pi h = 4.5\pi
   $$
   $$
   0.75h = 4.5 \quad \Rightarrow \quad h = \frac{4.5}{0.75} = 6 \text{ inches}
   $$

**Conclusion:**
To contain all of the ice cream inside the cone, the cone should be **6 inches tall**.

**Answer:**  
A cone 6 inches tall will hold all of the ice cream.
A 6-inch-tall cone will hold all the ice cream.

7. A scoop of ice cream has a 3 -inch diameter. How tall should the ice cream cone of the same diameter be in order to contain all of the ice cream inside the cone?

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