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Home » Solution: To solve for \( h(x^2 - 2) \), start with the given function: - IX Labs

Solution: To solve for \( h(x^2 - 2) \), start with the given function: - IX Labs

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 Solution: The diagonal of the rectangle is the diameter of the circumscribed circle. Using the Pythagorean theorem, the diagonal $ d = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26 $ km. The circumference $ C = \pi \times d = 26\pi $ km. Thus, the circumference is $ \boxed{26\pi} $.

📌 Question: An online student studying STEM subjects is tasked with determining the ratio of the area of the incircle to the area of a right triangle with hypotenuse $ z $ and inradius $ c $. What is this ratio?

📌 Solution: The area of the incircle is $ \pi c^2 $. For a right triangle, the semiperimeter $ s = \frac{a + b + z}{2} $, and the inradius $ c = \frac{a + b - z}{2} $. The area of the triangle is $ r \times s = c \times (z + c) $. Therefore, the ratio is $ \frac{\pi c^2}{c(z + c)} = \frac{\pi c}{z + c} $. The answer is $ \boxed{\dfrac{\pi c}{z + c}} $.

📌 Question: A robotics researcher designs a circular obstacle avoidance path for a robot. If the robot's shortest detour forms a triangle with sides of 7 units, 24 units, and 25 units, what is the radius of the circumscribed circle around its path?

📌 Solution: For a right triangle with legs 7 and 24, and hypotenuse 25, the hypotenuse is the diameter of the circumscribed circle. The radius $ r = \frac{25}{2} = 12.5 $ units. Thus, the radius is $ \boxed{12.5} $.

📌 Question: A geographer uses GIS to analyze a rectangular region with a perimeter of 40 km and a length-to-width ratio of 3:2. What is the circumference of the circle that circumscribes this rectangle?

📌 Solution: Let the length be $ 3x $ and width $ 2x $. The perimeter $ 2(3x + 2x) = 10x =

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To solve this, we first recognize that the center of a regular pentago...
R = \frac{s}{2 \sin(\pi/5)}
The sine of \( \pi/5 \) is \( \sin(\pi/5) = \sin(36^\circ) \), which i...
R = \frac{s}{2 \times 0.5878} \approx \frac{s}{1.1756} \approx 0.8507s
Since there are 5 participants, each moving from a vertex to the cente...
\times R = 5 \times 0.8507s = 4.2535s
Thus, the total distance traveled by all participants is:
\boxed{4.2535s}
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