Question

Some scientists believe that the average surface temperature of the world has been rising steadily. The average surface temperature can bemodeled by where Tis temperature in${}^{\circ }c$ and tis years since 1950.

$T=0.02t+15.0$

(a)What do the slope and T-intercept represent?

(b)Use the equation to predict the average global surface temperature in 2050.

Short answer

(a) The slope of the $t-$intercept represents $\mathrm{0.02.}$

(b) The global surface temperature in $2050$is${17.0}^{\circ }.$

Step-by-step solution

Part a. Step 1. Concept of Slope.

Suppose a non-vertical line has a slope mand y-intercept b. This means that the line intersects the y-axis at the point $\left(0,b\right),$so the point slope form of the equation of the line with $x=0$and$y=b$ becomes

$\begin{array}{c}(y-b)=m(x-0)\\ y=mx+b\end{array}$

Part a. Step 2. Concept of y-intercept.

The y-coordinate is a point where a line curve or a surface intersects the y-axis.

Part a. Step 3. Determining the slope.

The given equation is

$T=0.02t+15.0$

where T is temperature in ${}^{\circ }c$ and t is year 1950.

The slope of T-intercept represented is$\mathrm{0.02.}$

Part b. Step 1. Concept of Slope.

Suppose a non-vertical line has a slope m and y-intercept b. This means that the line intersects the y-axis at the point $\left(0,b\right),$so the point slope form of the equation of the line with $x=0$and $y=b$becomes

$\begin{array}{c}(y-b)=m(x-0)\\ y=mx+b\end{array}$

Part b. Step 2. Concept to determine two points.

If the value of x is given, then we have to evaluate the y by substituting the value in the equation and solve for y.

Part b. Step 3. Predicting the average global surface temperature in 2050.

The given equation is

$T=0.02t+15.0$

where T is temperature in ${}^{\circ }c$ and t is year 1950.

In 2050,

$\begin{array}{l}t=2050-1950\\ t=100\end{array}$

Substituting $t=100,$we get

$\begin{array}{l}T=0.02\left(100\right)+15.0\\ T=2.0+15.0\\ T=17.0\end{array}$

Hence, the global surface temperature in 2050 is ${17.0}^{\circ }.$