Math Problem Statement

If 

w ′(t)

 is the rate of growth of a child in pounds per year, what does 

11

8

w ′(tdt

    represent?

The child's initial weight at birth.The child's weight at age 11.    The change in the child's weight (in pounds) between the ages of 8 and 11.The change in the child's age (in years) between the ages of 8 and 11.The child's weight at age 8.

Solution

The expression

811w(t)dt\int_{8}^{11} w'(t) \, dt

represents the change in the child's weight (in pounds) between the ages of 8 and 11.

Explanation:

  1. The integral of a rate of change, w(t)w'(t), over a specific interval [a,b][a, b] gives the total change in the quantity w(t)w(t) over that interval.
  2. Here, w(t)w'(t) is the rate of growth in pounds per year, and the integral from t=8t=8 to t=11t=11 sums up all the changes in the child's weight over this period.
  3. Therefore, the result of this integral gives the total change in the child's weight between ages 8 and 11.

Correct Answer:

The change in the child's weight (in pounds) between the ages of 8 and 11.


Would you like further details or have any questions? Here are 5 related questions for you to explore:

  1. What is the physical interpretation of a definite integral in terms of real-world problems?
  2. How would the interpretation change if w(t)w'(t) were a negative value over part of the interval?
  3. What if the integral limits were reversed, such as 118w(t)dt\int_{11}^{8} w'(t) \, dt?
  4. How would you calculate w(11)w(11) if you were given w(8)w(8) and 811w(t)dt\int_{8}^{11} w'(t) \, dt?
  5. Can you explain the connection between the Fundamental Theorem of Calculus and this problem?

Tip: Always interpret the units of the rate and the limits of integration to understand the physical meaning of an integral.

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Math Problem Analysis

Mathematical Concepts

Definite Integration
Rate of Change
Physical Interpretation of Integrals

Formulas

∫ₐᵇ f'(x) dx = f(b) - f(a)

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12