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Solve all that

創建於：2025年5月9日

使用 O3 回答 Chat01

問題

IMG_5295.jpegIMG_5295.jpeg
IMG_5294.pngIMG_5294.png
IMG_5293.jpegIMG_5293.jpeg
IMG_5292.jpegIMG_5292.jpeg

Solve all that

Python

python
from PIL import Image
import matplotlib.pyplot as plt

img = Image.open('/mnt/data/IMG_5295.jpeg')
plt.imshow(img); plt.axis('off')

(-0.5, 1169.5, 1463.5, -0.5)

Python

python
plt.figure(figsize=(4,8))
plt.imshow(img.crop((580,100,720,350))); plt.axis('off')

(-0.5, 139.5, 249.5, -0.5)

Python

python
plt.imshow(img.crop((480,80,730,380))); plt.axis('off')

(-0.5, 249.5, 299.5, -0.5)

Python

python
plt.imshow(img.crop((350,0,760,450))); plt.axis('off')

(-0.5, 409.5, 449.5, -0.5)

Python

python
plt.imshow(img.crop((40,320,340,580))); plt.axis('off')

(-0.5, 299.5, 259.5, -0.5)

Python

python
plt.imshow(img.crop((120,400,500,780))); plt.axis('off')

(-0.5, 379.5, 379.5, -0.5)

Python

python
img2 = Image.open('/mnt/data/IMG_5294.png')
plt.imshow(img2); plt.axis('off')

(-0.5, 1169.5, 2531.5, -0.5)

Python

python
plt.imshow(img2.crop((0,0,460,780))); plt.axis('off')

(-0.5, 459.5, 779.5, -0.5)

Python

python
plt.imshow(img2.crop((0,0,640,1000))); plt.axis('off')

(-0.5, 639.5, 999.5, -0.5)

Python

python
plt.imshow(img2.crop((0,900,640,1700))); plt.axis('off')

(-0.5, 639.5, 799.5, -0.5)

Python

python
plt.imshow(img2.crop((0,850,640,1200))); plt.axis('off')

(-0.5, 639.5, 349.5, -0.5)

Python

python
plt.imshow(img2.crop((0,1150,640,2000))); plt.axis('off')

(-0.5, 639.5, 849.5, -0.5)

Python

python
plt.imshow(img2.crop((0,1700,640,2600))); plt.axis('off')

(-0.5, 639.5, 899.5, -0.5)

Python

python
img3 = Image.open('/mnt/data/IMG_5293.jpeg')
plt.imshow(img3); plt.axis('off')

(-0.5, 1169.5, 1579.5, -0.5)

Python

python
img4 = Image.open('/mnt/data/IMG_5292.jpeg')
plt.imshow(img4); plt.axis('off')

(-0.5, 1169.5, 2410.5, -0.5)

Python

python
plt.imshow(img2.crop((360,60,620,380))); plt.axis('off')

(-0.5, 259.5, 319.5, -0.5)

Python

python
plt.imshow(img2.crop((340,80,480,340))); plt.axis('off')

(-0.5, 139.5, 259.5, -0.5)

Python

python
plt.imshow(img2.crop((440,60,680,380))); plt.axis('off')

(-0.5, 239.5, 319.5, -0.5)

Python

python
plt.imshow(img2.crop((450,50,720,530))); plt.axis('off')

(-0.5, 269.5, 479.5, -0.5)

Python

python
plt.figure(figsize=(2,4))
plt.imshow(img2.crop((620,0,1000,780))); plt.axis('off')

(-0.5, 379.5, 779.5, -0.5)

Python

python
import math, pprint, json, os, sys

105.29957264870546

Python

python
plt.imshow(img3.crop((80,410,240,700))); plt.axis('off')

(-0.5, 159.5, 289.5, -0.5)

Python

python
plt.imshow(img3.crop((450,700,700,1000))); plt.axis('off')

(-0.5, 249.5, 299.5, -0.5)

Python

python
plt.imshow(img3.crop((450,830,730,1250))); plt.axis('off')

(-0.5, 279.5, 419.5, -0.5)

Python

python
plt.imshow(img3.crop((450,850,1000,1400))); plt.axis('off')

(-0.5, 549.5, 549.5, -0.5)

Python

python
plt.imshow(img3.crop((190,950,440,1400))); plt.axis('off')

(-0.5, 249.5, 449.5, -0.5)

Python

python
plt.imshow(img3.crop((190,1150,440,1550))); plt.axis('off')

(-0.5, 249.5, 399.5, -0.5)

Python

python
plt.imshow(img3.crop((80,880,450,1240))); plt.axis('off')

(-0.5, 369.5, 359.5, -0.5)

Python

python
plt.imshow(img3.crop((80,650,450,880))); plt.axis('off')

(-0.5, 369.5, 229.5, -0.5)

Python

python
plt.imshow(img3.crop((350,700,730,1100))); plt.axis('off')

(-0.5, 379.5, 399.5, -0.5)

Python

python
vol3 = (1/3)*3.14*(3.1**2)*5.9
vol3

59.34495333333334

Python

python
vol4 = (1/3)*3.14*(12**2)*9
vol4

1356.48

Python

python
plt.imshow(img4.crop((0,300,640,600))); plt.axis('off')

(-0.5, 639.5, 299.5, -0.5)

Python

python
plt.imshow(img4.crop((300,60,700,350))); plt.axis('off')

(-0.5, 399.5, 289.5, -0.5)

Python

python
plt.imshow(img4.crop((40,810,480,1400))); plt.axis('off')

(-0.5, 439.5, 589.5, -0.5)

Python

python
plt.imshow(img4.crop((40,1200,480,1700))); plt.axis('off')

(-0.5, 439.5, 499.5, -0.5)

Python

python
plt.imshow(img4.crop((40,1600,480,2200))); plt.axis('off')

(-0.5, 439.5, 599.5, -0.5)

Python

python
plt.imshow(img4.crop((40,2200,480,3000))); plt.axis('off')

(-0.5, 439.5, 799.5, -0.5)

Python

python
plt.imshow(img4.crop((40,1100,440,1600))); plt.axis('off')

(-0.5, 399.5, 499.5, -0.5)

Python

python
plt.imshow(img4.crop((40,1400,440,2200))); plt.axis('off')

(-0.5, 399.5, 799.5, -0.5)

Python

python
x,y=sp.symbols('x y')
sol=sp.solve([-3*x+5*y-21,2*x-3*y+13],[x,y])
sol

{x: -2, y: 3}

Python

python
vol13=3.14*(3.4**2)*1.2
vol13

43.55808

Python

python
plt.imshow(img4.crop((380,1150,900,1600))); plt.axis('off')

(-0.5, 519.5, 449.5, -0.5)

Python

python
plt.imshow(img4.crop((600,1150,1000,1600))); plt.axis('off')

(-0.5, 399.5, 449.5, -0.5)

Python

python
vol8 = (1/3)*3.14*(16.4**2)*12.9
vol8

3631.49792

Python

python
vol5=(1/3)*3.14*(9**2)*15
vol5

1271.7

Python

python
3.14*1215

3815.1000000000004

Python

python
straight = math.sqrt(78**2 + 13**2)
straight

79.07591289387686

Python

python
plt.imshow(img4.crop((150,1600,440,2000))); plt.axis('off')

(-0.5, 289.5, 399.5, -0.5)

思考

For problem 8, I computed various radius options and focused on adjusting the dimensions for an accurate diagonal. By testing different radius and width combinations, I arrived at a solution where r=27 and w=36, which gives an exact diagonal of 45. The other combinations either didn’t reach 45 or weren't the most suitable. Now, I'm finalizing test prep with the radius choices for verification, possibly providing a more typical example.

Thought for 11m 1s

回答

Here are complete step-by-step solutions (rounded as the book asks) for every numbered exercise that appears in the four photos you shared.

Page 386

#WorkAnswer
11 AHeight of a right-circular cone is found from h=s2−r2h=\sqrt{s^{2}-r^{2}}.<br>• Cone A: h=172−102=189≈13.75 cmh=\sqrt{17^{2}-10^{2}}=\sqrt{189}\approx13.75\text{ cm}<br>• Cone B: h=182−122=180≈13.42 cmh=\sqrt{18^{2}-12^{2}}=\sqrt{180}\approx13.42\text{ cm}Cone A is taller.
11 B13.75−13.42=0.33 cm13.75-13.42=0.33\text{ cm}0.33 cm
12In the triangular gate brace, x=382−242=868≈29.5 in.x=\sqrt{38^{2}-24^{2}}=\sqrt{868}\approx29.5\text{ in.}29.5 in.
13Space diagonal of a cube d=s3=23≈3.5 md=s\sqrt3=2\sqrt3\approx3.5\text{ m}3.5 m

Page 384 – On Your Own

#WorkAnswer
3 ASidewalk: 78+13=91 m78+13=91\text{ m}; grass shortcut: 782+132≈79.1 m\sqrt{78^{2}+13^{2}}\approx79.1\text{ m}Sidewalk route
3 B91−79.1≈11.9 m91-79.1\approx11.9\text{ m}11.9 m
4Space diagonal =362+242+962≈105.3 in.=\sqrt{36^{2}+24^{2}+96^{2}}\approx105.3\text{ in.}105.3 in.
5Slant height x=102+42=116≈10.77 mx=\sqrt{10^{2}+4^{2}}=\sqrt{116}\approx10.77\text{ m}10.77 m
6Need any l,wl,w with l2+w2≈45\sqrt{l^{2}+w^{2}}\approx45. Two convenient pairs:<br>• 27.0 cm×36.0 cm27.0\text{ cm}\times36.0\text{ cm} (exact 3-4-5 triple)<br>• 17.3 cm×41.5 cm17.3\text{ cm}\times41.5\text{ cm} (rounded)Any two correct pairs; sample shown.
7Face diagonal of 24 in×16 in24\text{ in}\times16\text{ in} rectangle: x=242+162≈28.8 in.x=\sqrt{24^{2}+16^{2}}\approx28.8\text{ in.}28.8 in.
8In the cone, r=s2−h2=902−402=6500≈80.6 mmr=\sqrt{s^{2}-h^{2}}=\sqrt{90^{2}-40^{2}}=\sqrt{6500}\approx80.6\text{ mm}80.6 mm
9Space diagonal =542+122+402≈68.3 cm=\sqrt{54^{2}+12^{2}+40^{2}}\approx68.3\text{ cm}68.3 cm

Pages 444–445 – Find Volume of Cones

Use V=13πr2hV=\dfrac{1}{3}\pi r^{2}h with π≈3.14\pi\approx3.14 where asked.

#WorkVolume (nearest 0.1)
1A cylinder with same rr and hh as a cone has 3 × the cone’s volume.<br>Given cone fountain V=2.6 m3V=2.6\text{ m}^3 ⇒ cylinder V=7.8 m3V=7.8\text{ m}^3.7.8 m^3
2r=2.0×103 mm,  h=6.0×103 mmr=2.0\times10^{3}\text{ mm},\;h=6.0\times10^{3}\text{ mm}<br>V=13πr2h=13π(2.0×103)2(6.0×103)=8.0×109π mm3V=\dfrac{1}{3}\pi r^{2}h=\dfrac{1}{3}\pi(2.0\times10^{3})^{2}(6.0\times10^{3})=8.0\times10^{9}\pi\text{ mm}^38.0×109π mm38.0\times10^{9}\pi\text{ mm}^3
3r=3.1 cm,  h=5.9 cmr=3.1\text{ cm},\;h=5.9\text{ cm}59.3 cm^3
4r=12 in,  h=9 inr=12\text{ in},\;h=9\text{ in}1 356.5 in^3
5r=9 ft,  h=15 ftr=9\text{ ft},\;h=15\text{ ft}1 271.7 ft^3
6r=5 m,  h=3 mr=5\text{ m},\;h=3\text{ m}78.5 m^3
7With diameter dd (so r=d2r=\dfrac{d}{2}):<br>V=13π(d2)2h=πd2h12V=\dfrac{1}{3}\pi\left(\dfrac{d}{2}\right)^{2}h=\boxed{\dfrac{\pi d^{2}h}{12}}

Page 445 – Test Prep

#Solution
8Substitute into V=13πr2hV=\frac{1}{3}\pi r^{2}h: V=13π(16.4)2(12.9) cm3\displaystyle V=\frac{1}{3}\pi(16.4)^{2}(12.9)\text{ cm}^{3} (≈ 3 631.5 cm^3). The four boxes should contain 1, 3, 16.4, 12.9 in that order.
9For r=3 ft,h=5 ftr=3\text{ ft},h=5\text{ ft}:<br>• Cone volume =15π ft3=15\pi\text{ ft}^3<br>• Cylinder volume =45π ft3=45\pi\text{ ft}^3<br>True statements are the ones saying: volume of the cone is 15π15\pi; volume of the cylinder is 45π45\pi; the cylinder’s volume is 3 times that of the cone; the ratio (V_{\text{cone}}:V_{\text{cyl}}=1:3.
10Smallest V=13πr2hV=\dfrac{1}{3}\pi r^{2}h comes from the smallest r2hr^{2}h. Computing each choice shows radius 1 m, height 15 m gives the least volume.

Page 445 – Spiral Review

#WorkAnswer
11Slope 100−5030−10=2.5\dfrac{100-50}{30-10}=2.5.The line rises about 2.5 quarts of iced coffee for every 1 °F increase in temperature.
12Solve {−3x+5y=212x−3y=−13\begin{cases}-3x+5y=21\\[2pt]2x-3y=-13\end{cases} ⇒ x=−2,  y=3x=-2,\;y=3.(−2, 3)(-2,\,3)
13Cylinder: V=πr2h=3.14(3.4)2(1.2)≈43.6 m3V=\pi r^{2}h=3.14(3.4)^{2}(1.2)\approx43.6\text{ m}^3.43.6 m^3

Feel free to let me know if you’d like any extra explanation or alternative methods for any of these problems!

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