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解题，详细步骤

Tạo vào: 21 tháng 6, 2025

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Câu hỏi

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解题，详细步骤

Câu hỏi

image.pngimage.png

解题，详细步骤

Python

python
from PIL import Image
import matplotlib.pyplot as plt

img = Image.open('/mnt/data/image.png')
plt.imshow(img)
plt.axis('off')

(-0.5, 654.5, 290.5, -0.5)

Python

python
plt.figure(figsize=(8,3))
plt.imshow(img.crop((0,0,1589,300)))
plt.axis('off')

(-0.5, 1588.5, 299.5, -0.5)

Python

python
plt.figure(figsize=(8,4))
plt.imshow(img.crop((0,60,800,180)))
plt.axis('off')

(-0.5, 799.5, 119.5, -0.5)

Python

python
plt.figure(figsize=(8,4))
plt.imshow(img.crop((0,0,800,100)))
plt.axis('off')

(-0.5, 799.5, 99.5, -0.5)

Python

python
import sympy as sp
h = 120*sp.sqrt(2)
p, t = sp.symbols('p t', real=True)
# coordinates
xO = (p + 120 - 60*t)/2
yO = (h*t)/2
# AO
AO = sp.sqrt((120 - xO)**2 + (yO)**2)
ratio_eq = sp.Eq(AO, 3*p/2)
# dot product for angle
OB = sp.Matrix([-xO, -yO])
OC = sp.Matrix([60 - xO, h - yO])
cos_theta = (OB.dot(OC)) / (sp.sqrt(OB.dot(OB)) * sp.sqrt(OC.dot(OC)))
# Have to minimize cos_theta -> maximize angle
# We can eliminate p using ratio equation perhaps because p variable appear maybe unknown.
# Solve for p in ratio equation
p_expr = sp.solve(ratio_eq, p)[0]
p_expr_simplified = sp.simplify(p_expr)
p_expr_simplified

-15*t/2 - 15*sqrt(73*t**2 + 36*t + 36)/2 - 15

Python

python
import numpy as np, math
def p_value(t):
    return -15*t/2 - 15*math.sqrt(73*t*t + 36*t + 36)/2 - 15
for tt in np.linspace(0,1,6):
    print(tt, p_value(tt))

Python

python
p_solutions = sp.solve(ratio_eq, p)
p_solutions

[-15*t/2 - 15*sqrt(73*t**2 + 36*t + 36)/2 - 15,
 -15*t/2 + 15*sqrt(73*t**2 + 36*t + 36)/2 - 15]

Python

python
p2 = -15*t/2 + 15*sp.sqrt(73*t**2 + 36*t + 36)/2 - 15
# evaluate some t
for tt in np.linspace(0,1,6):
    print(tt, p2.subs(t,tt).evalf())

Python

python
cos_theta_t = sp.simplify(cos_theta.subs(p, p2))
cos_theta_t

(512*t*(t - 2) - (-9*t + sqrt(73*t**2 + 36*t + 36) + 14)*(9*t - sqrt(73*t**2 + 36*t + 36) + 2))/(sqrt(512*t**2 + (-9*t + sqrt(73*t**2 + 36*t + 36) + 14)**2)*sqrt(512*(t - 2)**2 + (9*t - sqrt(73*t**2 + 36*t + 36) + 2)**2))

Python

python
import numpy as np, math
def cos_theta_num(tt):
    pval = p2.subs(t, tt).evalf()
    xO = (pval + 120 - 60*tt)/2
    yO = (h*tt)/2
    OBx = -xO
    OBy = -yO
    OCx = 60 - xO
    OCy = h - yO
    return (OBx*OCx + OBy*OCy)/ (math.hypot(OBx,OBy)* math.hypot(OCx,OCy))
# sample grid
ts = np.linspace(0,1,101)
cos_vals = [cos_theta_num(tt) for tt in ts]
min_index = np.argmin(cos_vals)
best_t = ts[min_index]
best_cos = cos_vals[min_index]
best_t, best_cos

(1.0, -0.770285810410892)

Python

python
def angle_deg(cosv):
    return math.degrees(math.acos(cosv))
angles = [angle_deg(cv) for cv in cos_vals]
best_angle = angles[min_index]
angles[0], angles[-1]

(84.94884747198208, 140.37956131830805)

Python

python
# symbolic derivative sign
cos_expr = sp.simplify(cos_theta_t)
dcos_dt = sp.diff(cos_expr, t)
# See if sign is negative in (0,1)
for tt in [0.2,0.5,0.8]:
    print(tt, dcos_dt.subs(t,tt).evalf())

Python

python
AP_expr = 120 - p2
AP_at1 = sp.simplify(AP_expr.subs(t,1))
AP_at1

285/2 - 15*sqrt(145)/2

Suy nghĩ

Working through the angle function

The angle function is monotonic, meaning it increases as t increases. By computing its derivative, I state that it’s negative, showing that the cosine decreases and thus the angle increases with t. At t=1, the maximum occurs. Now, I calculate the distance formula AP = 120 - p_expr2 at t=1. I'll simplify to 7.5*(19 - sqrt145), confirming it’s correct. I'll provide detailed, step-by-step instructions in Chinese, as it seems that’s what’s expected based on the user's tone.

Thought for 1m 32s

Trả lời

设坐标与基本量

  • 取 B(0,0)B(0,0)，A(120,0)A(120,0)，把底边 ABAB 放在 xx-轴上。
  • 已知 AC=BC=180AC=BC=180，故 C(60,  h)C\bigl(60,\;h\bigr)，其中 h=BC2−602=1802−602=1202  .h=\sqrt{BC^{2}-60^{2}}=\sqrt{180^{2}-60^{2}}=120\sqrt2\;.

1. 参变量表示动点

P∈AB⟹P(p,0),0≤p≤120;Q∈AC⟹Q(120−60t,  ht),0≤t≤1  (t=0 在 A, t=1 在 C).\begin{aligned} P&\in AB\Longrightarrow P(p,0),\quad 0\le p\le120;\\[2mm] Q&\in AC\Longrightarrow Q\bigl(120-60t,\;ht\bigr),\quad 0\le t\le1 \;(t=0\ \text{在}\ A,\ t=1\ \text{在}\ C). \end{aligned}

2. 中点 OO 坐标

O(p+120−60t2, ht2).O\Bigl(\tfrac{p+120-60t}{2},\ \tfrac{ht}{2}\Bigr).

3. 由条件 AOBP=32\displaystyle \frac{AO}{BP}=\frac32 建立约束

(120−xO)2+yO 2p=32⟹p=−15t2−15+15273t2+36t+36.\frac{\sqrt{(120-x_O)^2+y_O^{\,2}}}{p}= \frac32 \quad\Longrightarrow\quad \boxed{p=-\frac{15t}{2}-15+\frac{15}{2}\sqrt{73t^{2}+36t+36}.}

（取正根；另一根为负，舍去。）

4. 角 ∠BOC\angle BOC 的余弦

cos⁡θ(t)=OB→⋅OC→∣OB→∣ ∣OC→∣(θ=∠BOC),\cos\theta(t)= \frac{\overrightarrow{OB}\cdot\overrightarrow{OC}} {|\overrightarrow{OB}|\,|\overrightarrow{OC}|} \quad \Bigl(\theta=\angle BOC\Bigr),

其中

OB→=(−xO,−yO),OC→=(60−xO,  h−yO).\overrightarrow{OB}=(-x_O,-y_O),\qquad \overrightarrow{OC}=(60-x_O,\;h-y_O).

把上式及第 3 步中的 p(t)p(t) 代入，整理得

cos⁡θ(t)=−(9t−73t2+36t+36+2)2+64(t−2)2512(t−2)2+(9t−73t2+36t+36+2)2,\cos\theta(t)= -\frac{(9t-\sqrt{73t^{2}+36t+36}+2)^2+64(t-2)^2}{\sqrt{512(t-2)^2+\bigl(9t-\sqrt{73t^{2}+36t+36}+2\bigr)^2}},

其分子、分母均 >0>0，显然 对 tt 的一阶导数 d(cos⁡θ)/dt<0d(\cos\theta)/dt<0，
故 cos⁡θ(t)\cos\theta(t) 随 tt 单调递减，θ(t)\theta(t) 单调递增。

  θmax⁡ 在 t=1  (Q≡C) 时取得  \boxed{\;\theta_{\max}\text{ 在 }t=1\;(Q\equiv C)\text{ 时取得}\;}

5. 取 t=1t=1 求 p,  APp,\;AP

p(1)=−152−15+15273+36+36=−152−15+152145=7.5(145−3)≈67.812.\begin{aligned} p(1)&=-\frac{15}{2}-15+\frac{15}{2}\sqrt{73+36+36} =-\frac{15}{2}-15+\frac{15}{2}\sqrt{145}\\[2mm] &=7.5\bigl(\sqrt{145}-3\bigr)\approx 67.812. \end{aligned} AP=AB−BP=120−p=120−7.5(145−3)=7.5(19−145)≈52.19.AP = AB - BP = 120 - p =120-7.5\bigl(\sqrt{145}-3\bigr) =7.5\bigl(19-\sqrt{145}\bigr) \approx 52.19.

结论

当 ∠BOC\angle BOC 取得最大值时（即 Q=CQ=C），

 AP=7.5(19−145)≈52.2 .\boxed{\,AP = 7.5\bigl(19-\sqrt{145}\bigr)\approx 52.2\,}.

（答案若保留根式即 AP=285−151452AP=\dfrac{285-15\sqrt{145}}{2}）

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