answer:1. **Setup**: Consider two circles, one with radius R and the other with radius r, where R > r. Both circles are externally tangent to each other. We need to find the length of the external tangent between them, given that the tangents between the points of tangency form a right angle. 2. **Geometric Construction**: Consider the points of tangency on each circle where the external tangents touch. Let these points be A and B on the circle with radius R, and C and D on the circle with radius r, where A is tangent to C and B to D. 3. **Square Properties**: For each of the circles, the quadrilateral formed by the radii at the points of tangency and the tangents at these points is a square. This is because the radii are perpendicular to the tangents and both tangents at each point create right angles with the radii. 4. **Square Sides**: The side lengths of these squares are the radii R and r, respectively. 5. **Tangent Length Calculation**: - Since the tangents create a right angle, consider the entire setup as forming two right-angled triangles joined back-to-back with hypotenuses equal to R + r (distance between the centers of the circles). - The length L of each external tangent (the part we wish to find) can be related by the Pythagorean theorem, considering the total distance between the points of tangency. 6. **Applying Pythagorean Theorem**: - Using Pythagorean theorem directly between the distance between the circle centers, the external tangents, and the hypotenuses of the squares: [ L = sqrt{(R + r)^2 - (R - r)^2} ] - Simplifying this: [ L = sqrt{(R + r)^2 - (R - r)^2} = sqrt{(R^2 + 2Rr + r^2) - (R^2 - 2Rr + r^2)} = sqrt{4Rr} = 2sqrt{Rr} ] 7. **Resulting Length**: Therefore, the length of the external tangents between the circles can be given by: [ L = 2sqrt{Rr} ] # Conclusion: [ boxed{2sqrt{Rr}} ]

answer:Each number from 1 to 8 on the die has a probability of (frac{1}{8}) of being rolled. The expected value, (E), is calculated by: [ E = left(frac{1}{8} times 1^3right) + left(frac{1}{8} times 2^3right) + left(frac{1}{8} times 3^3right) + left(frac{1}{8} times 4^3right) + left(frac{1}{8} times 5^3right) + left(frac{1}{8} times 6^3right) + left(frac{1}{8} times 7^3right) + left(frac{1}{8} times 8^3right) ] [ E = frac{1}{8}(1 + 8 + 27 + 64 + 125 + 216 + 343 + 512) ] [ E = frac{1}{8}(1296) = 162 ] Rounding to the nearest cent, the expected value is (boxed{162.00}).

answer:Using the identity gcd(a,b) cdot mathop{text{lcm}}[a,b] = ab, we have: mathop{text{lcm}}[a,b] = frac{ab}{5}. a and b must be 4-digit multiples of 5, so our choices for each are: 1005, 1010, 1015, 1020, 1025, ldots To minimize ab, and thereby minimize mathop{text{lcm}}[a,b], we choose the smallest possible values for a and b that are not equal to ensure gcd(a, b) = 5. Selecting a = 1005 and b = 1010, we verify gcd(1005, 1010) = 5 as desired. Therefore, the smallest possible value for the least common multiple is: begin{align*} mathop{text{lcm}}[1005,1010] &= frac{1005 cdot 1010}{5} &= 1005 cdot 202 &= (1000 cdot 202) + (5 cdot 202) &= 202{,}000 + 1010 &= boxed{203{,}010}. end{align*}

answer:Let's denote the constant rate of the slower pump as S and the constant rate of the faster pump as F. We know that the slower pump would take 12.5 hours to fill the pool alone, so its rate is 1 pool per 12.5 hours, or S = 1/12.5 pools per hour. When both pumps work together, they fill the pool in 5 hours. The combined rate of both pumps is 1 pool per 5 hours, or (S + F) = 1/5 pools per hour. We can express F in terms of S: F = (1/5) - S. Since we know S = 1/12.5, we can substitute this into the equation for F: F = (1/5) - (1/12.5). To subtract these fractions, we need a common denominator, which would be 5 * 12.5 = 62.5. So we convert each fraction to have a denominator of 62.5: F = (12.5/62.5) - (5/62.5) F = (12.5 - 5) / 62.5 F = 7.5 / 62.5 Now we have the rates of both pumps in terms of the same denominator, we can find the ratio of the faster pump's rate to the slower pump's rate: Ratio (F:S) = (7.5 / 62.5) : (1 / 12.5) Ratio (F:S) = (7.5 / 62.5) * (12.5 / 1) (since multiplying by the reciprocal is the same as dividing) Ratio (F:S) = (7.5 * 12.5) / (62.5 * 1) Ratio (F:S) = 93.75 / 62.5 Ratio (F:S) = 1.5 So the ratio of the constant rate of the faster pump to the constant rate of the slower pump is boxed{1.5:1} .