Question

Write an exponential model given the two points (6,140) and (7,220).

Options:
a) y = 140 \times 1.2^xy=140×1.2^x

b) y = 140 \times 1.4^xy=140×1.4^x

c) y = 220 \times 1.2^xy=220×1.2^x

d) y = 220 \times 1.4^xy=220×1.4^x

Answer 1

The exponential model given two points (6, 140) and (7, 220) is represented by option (b), which is y = 140 × 1.4^x, after calculating the base growth rate from the given points.

Step-by-step explanation:

To write an exponential model given two points (6, 140) and (7, 220), you must first determine the base growth rate. Let's denote the exponential model by y = a × b^x, where a is the initial amount and b is the base of the exponential function, representing the growth rate.

Using the first point (6, 140), you can see that the initial amount a must be 140 because that's the value of y when x is 6. To find the growth rate b, use the second point (7, 220) and the fact that when x increased by 1, y went from 140 to 220. This gives you an equation 140 × b^1 = 220.

To solve for b, divide both sides by 140, resulting in b = 220 / 140 = 1.5714. However, since the options given to us do not contain this exact number, we must select the closest option.

Comparing our result with the answer options, b is closer to 1.4 than to 1.2, making option (b) y = 140 × 1.4^x the correct exponential model. Options (a), (c), and (d) can be eliminated because they do not match our calculated base or initial amount.