https://spherical-harmonics.hateblo.jp/entry/Kyodai/1923/Kougaku

2025.01.13記

[1] Four equal balls are included in a sphere of radius $R$ . Calculate maximum diameter of the balls.

[2] (a) Find the value of $\dfrac{a^{\sin x}-a}{\log\sin x}$ when $x=\dfrac{\pi}{2}$ .

(b) FInd the value of $\theta$ which makes $\dfrac{\sin\theta\cos\theta}{\cos^2\left(\dfrac{\pi}{3}-\theta\right)}$ maximum.

(d) Change the independent variable from $x$ to $z$ in $x^2\dfrac{d^2y}{dx^2}+2x\dfrac{dy}{dx}+\dfrac{a^2}{x^2}y=0$ when $x=\dfrac{1}{z}$ .

(3) (a) Calculate the curve length of one arch of a common cycloid, diameter of the rolling circle begin $2a$ .

(b) Calculate the surface area of a paraboloid of revolution which $10$ cm depth and $40$ cm width.