1Cademy - Interpreting the Slope and T-Intercept of $$T = \frac{1}{4}n + 40$$

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Slope-Intercept Form of an Equation of a Line

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Interpreting the Slope and T-Intercept of $T = \frac{1}{4}n + 40$

The mathematical equation $T = \frac{1}{4}n + 40$ translates as a modeled real-world system aimed at estimating ambient outside temperature in degrees Fahrenheit, $T$ , relying solely on the quantity of cricket chirps observed within a single minute, $n$ . Processing the formula through the slope–intercept lens proves that it maintains a slope of $\frac{1}{4}$ and originates from a $T$ -intercept of $(0, 40)$ .

ⓐ Estimate the temperature when there are absolutely no chirps: Substituting an $n$ input of $0$ generates: $T = \frac{1}{4}(0) + 40 = 0 + 40 = 40$ degrees.

ⓑ Estimate the temperature matching $100$ total chirps per minute: Substituting an $n$ input of $100$ generates: $T = \frac{1}{4}(100) + 40 = 25 + 40 = 65$ degrees.

ⓒ Interpret the slope and $T$ -intercept:

Possessing a slope modeled around $\frac{1}{4}$ conceptually implies that the outdoor environmental temperature ( $T$ ) increases consistently by $1$ degree Fahrenheit for each and every batch of $4$ independent cricket chirps ( $n$ ) acquired per minute.
An established $T$ -intercept placed at $(0, 40)$ demonstrates that if literally $0$ cricket chirps are output during a minute's duration, the calculated starting temperature equals precisely $40$ degrees Fahrenheit.

ⓓ Graph the equation: Since observational values run quite high, prepare a coordinate plane outfitted using a properly expanded numbering scale. Plot the initial mathematical reading at the $T$ -intercept $(0, 40)$ . Follow the embedded slope parameters by counting a positive rise ascending $1$ coordinate unit upward, aligned sequentially alongside a run pushing $4$ coordinate units right; establishing the secondary coordinate directly at $(4, 41)$ . Connecting both dots with a straight line constructs the graphical relationship.

Updated 2026-04-23

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OpenStax Intermediate Algebra

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