In 1993, there were 41,778 shopping centers in a certain country. In 2003 , there were 48,103 .
(a) Write an equation expressing the number of shopping centers in terms of the number of years after 1993 .
(b) When will the number of shopping centers reach 70,000 ?
(a) The equation is .
(Type integers or decimals.)
(b) The number of shopping centers will reach 70,000 in the year .
(Round down to the nearest year.)

Ask by Montgomery Gray. in the United States

Dec 15,2024

Question

In 1993, there were 41,778 shopping centers in a certain country. In 2003 , there were 48,103 .
(a) Write an equation expressing the number of shopping centers in terms of the number of years after 1993 .
(b) When will the number of shopping centers reach 70,000 ?
(a) The equation is .
(Type integers or decimals.)
(b) The number of shopping centers will reach 70,000 in the year .
(Round down to the nearest year.)

Ask by Montgomery Gray. in the United States

Dec 15,2024

UpStudy AI · Accepted Answer

To solve the problem, we will first determine the equation that expresses the number of shopping centers in terms of the number of years after 1993.

### (a) Finding the equation

1. **Identify the points**: 
   - In 1993 (when $$ x = 0 $$), there were 41,778 shopping centers. This gives us the point $$ (0, 41778) $$.
   - In 2003 (when $$ x = 10 $$), there were 48,103 shopping centers. This gives us the point $$ (10, 48103) $$.

2. **Calculate the slope (m)**:
   $$
   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{48103 - 41778}{10 - 0} = \frac{6325}{10} = 632.5
   $$

3. **Use the point-slope form to find the equation**:
   We can use the point $$ (0, 41778) $$ to write the equation in slope-intercept form $$ y = mx + b $$:
   $$
   y = 632.5x + 41778
   $$

Thus, the equation is:
$$
y = 632.5x + 41778
$$

### (b) Finding when the number of shopping centers will reach 70,000

1. **Set $$ y = 70000 $$** and solve for $$ x $$:
   $$
   70000 = 632.5x + 41778
   $$
   $$
   70000 - 41778 = 632.5x
   $$
   $$
   28222 = 632.5x
   $$
   $$
   x = \frac{28222}{632.5} \approx 44.6
   $$

2. **Determine the year**:
   Since $$ x $$ represents the number of years after 1993, we add $$ x $$ to 1993:
   $$
   1993 + 44.6 \approx 2037.6
   $$
   Rounding down to the nearest year gives us 2037.

### Final Answers

(a) The equation is $$ y = 632.5x + 41778 $$.

(b) The number of shopping centers will reach 70,000 in the year $$ 2037 $$.
(a) The equation is $$ y = 632.5x + 41778 $$. (b) The number of shopping centers will reach 70,000 in the year 2037.

Upstudy AI Solution

Answer

Solution

(a) Finding the equation

(b) Finding when the number of shopping centers will reach 70,000

Final Answers

Extra Insights

Related Questions

Latest Algebra Questions