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The cones are similar. Find the missing slant height
ℓ.

ℓ=◻yd

The cones are similar. Find the missing slant height $\ell$ . $\newline$ $\ell=\square \mathrm{yd}$

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Solve a quadratic equation using square roots

Full solution

Q. The cones are similar. Find the missing slant height

\ell

.

\newline

\ell=\square \mathrm{yd}

Identify Proportions: Identify the proportions of the similar cones. Let's say the dimensions of the smaller cone are $r_1$ , $h_1$ , and $\ell_1$ , and the dimensions of the larger cone are $r_2$ , $h_2$ , and $\ell_2$ . We know that the ratios of corresponding dimensions are equal because the cones are similar.
Set Up Proportion: Set up the proportion using the known values. $\newline$ Assume we have the values for $r_1$ , $h_1$ , $r_2$ , and $h_2$ , but $\ell_1$ is missing. The proportion is $\frac{r_1}{r_2} = \frac{h_1}{h_2} = \frac{\ell_1}{\ell_2}$ .
Plug in Values: Plug in the values and solve for $\ell_1$ . Let's say $r_1 = 3 \, \text{yd}$ , $r_2 = 6 \, \text{yd}$ , $h_1 = 4 \, \text{yd}$ , $h_2 = 8 \, \text{yd}$ , and $\ell_2 = 10 \, \text{yd}$ . So, $\frac{3}{6} = \frac{4}{8} = \frac{\ell_1}{10}.$
Simplify Ratios: Simplify the ratios and solve for $\ell_1$ . $\frac{3}{6}$ simplifies to $\frac{1}{2}$ , and $\frac{4}{8}$ also simplifies to $\frac{1}{2}$ . So, $\frac{1}{2} = \frac{\ell_1}{10}.$
Multiply by $10$ : Multiply both sides by $10$ to find $\ell_1$ . $\newline$ $\frac{1}{2} \times 10 = \ell_1$ .
Calculate: Calculate $\ell_1$ . $\newline$ $\ell_1 = 5 \, \text{yd}.$

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