Question 1185460: Table: Rate of the Cricket Chirps
Temperature in ℉
40
60
80
100
120
Rate
(Number of Chirps per Minute)
0
86
172
258
516
a.) Find a formula for g if g(t) represents the number of chirps per minute a cricket makes at temperature t degrees Fahrenheit.
b.) If f(c) represents the Fahrenheit reading that corresponds to a Celsius reading of c, which between the two functions g(f(t)) or f(g(t)) represents the number of chirps per minute a cricket makes when the temperature is c degrees Celsius?
c.) For the function in (b), write a formula for this and name it function h.
d.)Find the rate at which a cricket chirps if the temperature is __℉? __℃?
e.)Find the slope of the function g(t), h(c), and f(c). What does the slope of g(t) mean within the context of the problem?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this cricket chirp problem:
**(a) Formula for g(t):**
The data points suggest a linear relationship. Let's find the slope (m) and y-intercept (b) of the line.
1. **Choose two points:** Let's use (40, 0) and (60, 86).
2. **Calculate the slope (m):**
m = (change in y) / (change in x) = (86 - 0) / (60 - 40) = 86 / 20 = 4.3
3. **Find the y-intercept (b):** Use the point-slope form of a line (y - y₁ = m(x - x₁)) and one of the points, or plug one of the points into y=mx+b. Using (40,0):
0 = 4.3 * 40 + b
b = -172
4. **Write the equation:**
g(t) = 4.3t - 172
**(b) Which composite function represents chirps at Celsius temperature?**
* f(c) converts Celsius to Fahrenheit.
* g(t) converts Fahrenheit to chirps per minute.
Therefore, *g(f(c))* is the correct composite function. It first converts Celsius to Fahrenheit using f(c), and then converts the Fahrenheit temperature to chirps per minute using g(t).
**(c) Formula for h(c) = g(f(c)):**
1. **Formula for f(c):** The formula to convert Celsius to Fahrenheit is:
f(c) = (9/5)c + 32
2. **Substitute f(c) into g(t):**
h(c) = g(f(c)) = 4.3 * f(c) - 172
h(c) = 4.3 * ((9/5)c + 32) - 172
h(c) = (38.7/5)c + 137.6 - 172
h(c) = 7.74c - 34.4
**(d) Chirp rate at specific temperatures:**
You need to provide the specific temperatures (one in Fahrenheit and one in Celsius) to calculate the chirp rates. Plug the Fahrenheit temperature into g(t) and the Celsius temperature into h(c).
**(e) Slopes and their meaning:**
* **Slope of g(t):** The slope of g(t) is 4.3. This means that for every 1-degree Fahrenheit increase in temperature, the cricket chirp rate increases by 4.3 chirps per minute.
* **Slope of h(c):** The slope of h(c) is 7.74. This means that for every 1-degree Celsius increase in temperature, the cricket chirp rate increases by 7.74 chirps per minute.
* **Slope of f(c):** The slope of f(c) is 9/5 or 1.8. This means that for every 1-degree Celsius increase in temperature, the Fahrenheit temperature increases by 1.8 degrees.
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