교과서 질문
Spring work
b. It takes 50 N of force to stretch a spring 0.2 m from its equilibrium position. How much work is needed to stretch it an additional 0.5 m?
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Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
6장, 문제 6.R.77b
Pumping water A water tank has the shape of a box that is 2 m wide, 4 m long, and 6 m high.
b. If the water in the tank is 2 m deep, how much work is required to pump the water to a level of 1 m above the top of the tank?
검증된 단계별 안내1
First, identify the dimensions of the tank and the depth of the water. The tank is 2 m wide, 4 m long, and 6 m high. The water depth is 2 m.
Set up a coordinate system to represent the height of the water. Let \( y = 0 \) be at the bottom of the tank and \( y = 6 \) be at the top. The water occupies from \( y = 0 \) to \( y = 2 \).
Consider a thin horizontal slice of water at height \( y \) with thickness \( dy \). The volume of this slice is the area of the base times the thickness: \( dV = (2 \times 4) dy = 8 dy \) cubic meters.
Calculate the weight of this slice. Since the density of water is \( \rho = 1000 \, \text{kg/m}^3 \) and gravitational acceleration is \( g = 9.8 \, \text{m/s}^2 \), the weight is \( dW = \rho g dV = 1000 \times 9.8 \times 8 dy = 78400 dy \) newtons.
Determine the distance the slice must be lifted. The water must be pumped to 1 m above the top of the tank, so the lifting height for the slice at height \( y \) is \( (6 + 1) - y = 7 - y \) meters.
Set up the integral for the work done to pump all the water: \[ W = \int_0^2 \text{(weight of slice)} \times \text{(distance lifted)} \, dy = \int_0^2 78400 (7 - y) dy. \] This integral represents the total work required.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
7m주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Work Done by a Variable Force
Work is calculated as the integral of force over distance when the force varies. In pumping water, the force depends on the weight of each water layer, and the distance varies with depth, requiring integration to sum the work done on all layers.
추천 영상:
가이드 코스
05:40
Work Done On A Spring (Hooke's Law)
Volume and Weight of Water
The volume of water is found by multiplying the tank's cross-sectional area by the water depth. The weight is then calculated by multiplying the volume by the density of water and gravitational acceleration, which gives the force needed to move the water.
추천 영상:
04:48Finding Volume Using Disks
Setting up the Integral for Work
To find the total work, divide the water into thin horizontal slices, calculate the work to lift each slice to the target height, and integrate over the depth of the water. This involves expressing the distance each slice is lifted as a function of its depth.
추천 영상:
가이드 코스
06:22Introduction To Work
관련 실천
교과서 질문
Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:
b. Apply the shell method and integrate with respect to x.
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교과서 질문
43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.
The region bounded by the graphs of y = 2x,y = 6−x, and y = 0 is revolved about the line y = −2 and the line x = −2. Find the volumes of the resulting solids. Which one is greater?
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교과서 질문
An oscillator The acceleration of an object moving along a line is given by a(t) = 2 sin πt/4. The initial velocity and position are v(0)= −8/π and s(0)=0.
a. Find the velocity and position for t≥0.
22
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교과서 질문
58–61. Arc length Find the length of the following curves.
y = x³/6 + 1/2x on [1,2]
77
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교과서 질문
Lifting problem A 4-kg mass is attached to the bottom of a 5-m, 15-kg chain. If the chain hangs from a platform, how much work is required to pull the chain and the mass onto the platform?
68
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