这部分需要会什么？
- 确定变量及单位
- 将固定量和单位变化量写成线性关系
- 从总量、差量和单价关系建立方程或方程组
核心概念与公式
线性模型 \(y=mx+b\) 中，\(b\) 是 \(x=0\) 时的初始量，\(m\) 是每单位 \(x\) 带来的固定变化量。若 \(x\) 的单位为小时、\(y\) 为千米，\(m\) 的单位就是千米/小时。
建立模型的关键是找到关系：总价等于单价乘数量；路程等于速度乘时间；两类物品的总数等于两类数量之和。只有相同时间内增减量固定时，才适合直接用线性模型。
做题时怎样使用
- 用一句话定义每个变量，写明单位
- 把总量、固定费用、每单位费用等分别列出
- 建立等式并检查一个简单输入，例如数量为零时是否符合题意
容易混淆的地方
初始量未必是零。每次增长固定百分比通常是指数变化，不能直接写成每次增加固定数量。
用一个例题串起方法
知识点教学例题租车有 30 元固定费用，另按每千米 2.4 元收费。写出行驶 \(d\) 千米的费用 \(C\)，并求 18 千米的费用。
展开解题思路 ＋
固定费用是截距，单位里程费用是斜率，所以 \(C=30+2.4d\)。当 \(d=18\) 时，\(C=73.2\) 元。
本知识点题库
先独立作答，再查看答案与解析。
线性建模・建立方程挑战
A scientist is determining the rate of heat absorption in a material by measuring its temperature at regular time intervals. The initial temperature of the material is recorded as 15 degrees Celsius (°C). The temperature of the material increases at a constant rate of 3°C every 7 minutes for the first 70 minutes after the initial temperature was recorded. Which equation best represents the temperature y, in °C, of the material x minutes after the initial temperature was recorded, where \( 0\le x\le70 \)?
参考答案
线性建模・建立方程进阶
The cost to rent a bus from Company X is $950 for the first 2 hours and an additional $50 per hour for each hour after the first 2 hours. If the total cost to rent the bus from Company X for t hours, where t > 2, is $1,150, which equation represents this situation?
原库参考解析
线性建模・建立方程挑战
A beekeeper's initial observation of the population of a certain bee colony was 1,100 bees. The beekeeper set a goal to increase the population to 2,600 bees. The beekeeper uses a model that predicts the population of this bee colony begins at 1,100 and increases by 120 bees per week in the first two weeks after the initial observation, and then increases by 180 bees per week until the beekeeper's goal is reached. According to this model, at the end of week w after the initial observation, where \( w>2 \), which of the following functions gives the predicted number of bees still needed to reach the beekeeper's goal?
原库参考解析
线性建模・建立方程进阶
Sophia bought confetti and balloons for $69. Each package of confetti cost $2, and each balloon cost $1. If Sophia bought 10 packages of confetti, how many balloons did she buy?
原库参考解析
线性建模・建立方程挑战
Each 3.0-ounce serving of cheddar cheese and each 1.2-ounce serving of tuna provides about 1 microgram of vitamin B12. If a total of 3.8 micrograms of vitamin B12 are consumed from eating x ounces of cheese and y ounces of tuna, which equation best represents this situation?
原库参考解析
线性建模・建立方程进阶
The length of a stony coral colony was estimated to be 210 centimeters at the beginning of a study. The length of this coral colony was estimated to increase at a rate of 8 centimeters per year. What was the estimated length, in centimeters, of this coral colony 10 years after the study began?
原库参考解析
线性建模・建立方程进阶
In a soccer league, each team earns 1 point for each game the team plays that ends in a draw and 3 points for each game the team wins. A certain team in this league has earned 46 points. Which equation represents this situation, where d represents the number of games this team has played that ended in a draw and w represents the number of games this team has won?
原库参考解析
线性建模・建立方程挑战
Darrel has a blanket with an area of 110 square inches (in²). Darrel plans to enlarge the blanket by sewing on pieces of fabric, where each piece would increase the blanket area by 64 in². Which equation gives the blanket's total area, y , in square inches, after Darrel sews on x of these pieces of fabric?
原库参考解析
线性建模・建立方程基础
Jay walks at a speed of 3 miles per hour and runs at a speed of 5 miles per hour. He walks for w hours and runs for r hours for a combined total of 14 miles. Which equation represents this situation?
原库参考解析
线性建模・建立方程基础
Vivian bought party hats and cupcakes for $71. Each package of party hats cost $3, and each cupcake cost $1. If Vivian bought 10 packages of party hats, how many cupcakes did she buy?
原库参考解析
线性建模・建立方程挑战
An airplane descends from an altitude of 9,500 feet to 5,000 feet at a constant rate of 400 feet per minute. What type of function best models the relationship between the descending airplane's altitude and time?
原库参考解析
线性建模・建立方程挑战
A window repair specialist charges $220 for the first two hours of repair plus an hourly fee for each additional hour. The total cost for 5 hours of repair is $400. Which function \( f \) gives the total cost, in dollars, for x hours of repair, where \( x\ge2 \)?
原库参考解析
