Math Problem Statement

We can represent an elapsed time of 10.4 hours after the candle was lit. What is the change in time from 5.6 hours to 10.4 hours after the candle was lit, and what is the corresponding change in the candle's height at this time?

Solution

Let me guide you through the problem step by step based on the image provided:

Part (b): Change in time (Δt\Delta t)

You already answered this part correctly: Δt=10.45.6=4.8 hours\Delta t = 10.4 - 5.6 = 4.8 \text{ hours}

Part (c): Change in height (Δh\Delta h)

The rate at which the candle burns is given as 1.8 inches per hour. Therefore, the change in height is calculated using the formula: Δh=1.8×Δt\Delta h = -1.8 \times \Delta t Since Δt=4.8\Delta t = 4.8 hours: Δh=1.8×4.8=8.64 inches\Delta h = -1.8 \times 4.8 = -8.64 \text{ inches}

So, the change in height Δh=8.64\Delta h = -8.64 inches.

Part (d): Height after 10.4 hours

The initial height at t=5.6t = 5.6 hours is hi=9.92h_i = 9.92 inches (from the graph). Using the formula: h=hi+Δhh = h_i + \Delta h Substitute the values: h=9.92+(8.64)=1.28 inchesh = 9.92 + (-8.64) = 1.28 \text{ inches}

So, the height of the candle at 10.4 hours after it was lit is 1.28 inches.

Part (e): General formula for the candle height

The candle height decreases at a constant rate of 1.8 inches per hour. Given the height at t=0t = 0 as h0=18h_0 = 18 inches (from the graph), the general formula for the height hh at any time tt is: h=h01.8×th = h_0 - 1.8 \times t

This formula will give the height of the candle for any given tt.


Would you like to discuss any specific part in more detail?

Here are five related questions to practice:

  1. What would be the candle height at t=6t = 6 hours using the general formula?
  2. If the candle burns at 2 inches per hour, what would be the change in height over 5 hours?
  3. How would the formula for h(t)h(t) change if the initial height was 20 inches?
  4. What is the time tt when the candle completely burns out if the height starts at 18 inches?
  5. If the candle burned slower at 1 inch per hour, how long would it take to burn out?

Tip: Always track units during calculations, especially in rate problems involving time and physical dimensions!

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Math Problem Analysis

Mathematical Concepts

Time Calculations
Rate of Change
Linear Equations

Formulas

Change in time (Δt) = t_final - t_initial
Change in height (Δh) = rate * Δt
Height formula: h = h_i + Δh

Theorems

Linear Function Theorem

Suitable Grade Level

Grades 6-8