Step 1: Let $N = \sqrt[3]{\frac{75.4 \times 4.83^2}{0.00521}}$. To evaluate using logarithms, we take the logarithm of both sides: $$ \log N = \log \left( \left( \frac{75.4 \times 4.83^2}{0.00521} \right)^{\frac{1}{3}} \right) $$ $$ \log N = \frac{1}{3} \left[ \log(75.4) + \log(4.83^2) - \log(0.00521) \right] $$ $$ \log N = \frac{1}{3} \left[ \log(75.4) + 2\log(4.83) - \log(0.00521) \right] $$ Step 2: Find the logarithms of the numbers using a logarithm table (or calculator for precision): $$ \log(75.4) = 1.8774 $$ $$ \log(4.83) = 0.6839 $$ $$ 2\log(4.83) = 2 \times 0.6839 = 1.3678 $$ $$ \log(0.00521) = \bar{3}.7168 \quad (\text{which is } -3 + 0.7168) $$ Step 3: Substitute these values into the equation for $\log N$: $$ \log N = \frac{1}{3} [1.8774 + 1.3678 - (\bar{3}.7168)] $$ $$ \log N = \frac{1}{3} [1.8774 + 1.3678 - (-3 + 0.7168)] $$ $$ \log N = \frac{1}{3} [1.8774 + 1.3678 + 3 - 0.7168] $$ $$ \log N = \frac{1}{3} [3.2452 + 3 - 0.7168] $$ $$ \log N = \frac{1}{3} [6.2452 - 0.7168] $$ $$ \log N = \frac{1}{3} [5.5284] $$ $$ \log N = 1.8428 $$ Step 4: Find the antilogarithm of $1.8428$: $$ N = \text{antilog}(1.8428) $$ From antilog tables, the antilog of $0.8428$ is approximately $6.963$. Since the characteristic is $1$, the number has $1+1=2$ digits before the decimal point. $$ N = 69.63 $$ The final answer is $\boxed{69.63}$. --- Step 1: Start with the given formula: $$ a = \frac{bd}{Nb^2 - d} $$ Step 2: Multiply both sides by $(Nb^2 - d)$ to eliminate the denominator: $$ a(Nb^2 - d) = bd $$ Step 3: Distribute $a$ on the left side: $$ aNb^2 - ad = bd $$ Step 4: Collect all terms containing $d$ on one side of the equation. Add $ad$ to both sides: $$ aNb^2 = bd + ad $$ Step 5: Factor out $d$ from the terms on the right side: $$ aNb^2 = d(b + a) $$ Step 6: Isolate $d$ by dividing both sides by $(b + a)$: $$ d = \frac{aNb^2}{b + a} $$ The final answer is $\boxed{d = \frac{aNb^2}{b + a}}$. --- Step 1: Find the center of the circle. The center is the midpoint of the diameter PQ. Given points $P(-2, 2)$ and $Q(-2, -6)$. Let the center be $(h, k)$. $$ h = \frac{x_1 + x_2}{2} = \frac{-2 + (-2)}{2} = \frac{-4}{2} = -2 $$ $$ k = \frac{y_1 + y_2}{2} = \frac{2 + (-6)}{2} = \frac{-4}{2} = -2 $$ The center of the circle is $(-2, -2)$. Step 2: Find the radius of the circle. The radius is half the length of the diameter PQ. Length of diameter $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ $$ PQ = \sqrt{(-2 - (-2))^2 + (-6 - 2)^2} $$ $$ PQ = \sqrt{(0)^2 + (-8)^2} $$ $$ PQ = \sqrt{0 + 64} $$ $$ PQ = \sqrt{64} = 8 $$ The radius $r = \frac{PQ}{2} = \frac{8}{2} = 4$. Step 3: Write the equation of the circle in standard form $(x-h)^2 + (y-k)^2 = r^2$. Substitute the center $(-2, -2)$ and radius $r=4$: $$ (x - (-2))^2 + (y - (-2))^2 = 4^2 $$ $$ (x + 2)^2 + (y + 2)^2 = 16 $$ Step 4: Expand the equation to the general form $ax