We aim to determine the slant height $s$ of the frustum. Here's how we solve the problem step-by-step:

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1. Set Up Variables and Known Quantities

Let the dimensions of the original cone be:

• Radius of the base: $R$,
• Height: $H$,
• Slant height: $L$.

Let the dimensions of the smaller cone be:

• Radius of the base: $r$,
• Height: $h$,
• Slant height: $l$.

The original cone's volume is: $V_{\text{original}} = \frac{1}{3} \pi R^2 H = 4\pi \quad \text{(given)}.$ This implies: R^2 H = 12. \tag{1}

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2. Relation Between the Smaller Cone and the Frustum

The plane cutting the cone is parallel to its base, so the smaller cone is similar to the original cone. From similarity: $\frac{r}{R} = \frac{h}{H} = \frac{l}{L}.$

Let the ratio of similarity be $k = \frac{r}{R} = \frac{h}{H} = \frac{l}{L}$, where $0 < k < 1$.

The dimensions of the smaller cone become: r = kR, \quad h = kH, \quad l = kL. \tag{2}

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3. Volume of the Smaller Cone and Frustum

The volume of the smaller cone is: $V_{\text{smaller}} = \frac{1}{3} \pi r^2 h.$ Substitute $r = kR$ and $h = kH$: $V_{\text{smaller}} = \frac{1}{3} \pi (kR)^2 (kH) = \frac{1}{3} \pi k^3 R^2 H.$ Using $R^2 H = 12$ from equation (1): V_{\text{smaller}} = \frac{1}{3} \pi k^3 (12) = 4\pi k^3. \tag{3}

The volume of the frustum is: $V_{\text{frustum}} = V_{\text{original}} - V_{\text{smaller}}.$ Substitute the known volumes: V_{\text{frustum}} = 4\pi - 4\pi k^3 = 4\pi (1 - k^3). \tag{4}

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4. Condition on the Frustum’s Volume

The problem states that the volume of the frustum is $\frac{3}{4}$ of the original cone's volume: $V_{\text{frustum}} = \frac{3}{4} V_{\text{original}}.$ Substitute $V_{\text{original}} = 4\pi$: $4\pi (1 - k^3) = \frac{3}{4} (4\pi).$ Simplify: $4\pi (1 - k^3) = 3\pi.$ Divide through by $4\pi$: $1 - k^3 = \frac{3}{4}.$ Solve for $k^3$: k^3 = \frac{1}{4}. \tag{5}

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5. Slant Height of the Frustum

The slant height of the frustum is the difference in the slant heights of the original and smaller cones: $s = L - l.$ From similarity, $l = kL$, so: $s = L - kL = L(1 - k).$

To find $L$, note that the slant height of the original cone relates to its radius $R$ and height $H$: L = \sqrt{R^2 + H^2}. \tag{6}

Using $k = \sqrt[3]{\frac{1}{4}}$, calculate $1 - k$: $1 - k = 1 - \sqrt[3]{\frac{1}{4}}.$

Thus: $s = L \left( 1 - \sqrt[3]{\frac{1}{4}} \right).$

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6. Final Answer

Substitute $L$ explicitly if $R$ and $H$ are needed, but the result depends on the similarity ratio $k$ and the slant height $L$. You can stop here if additional numerical evaluation is not needed.