Answer:
Explanation:
To represent the sales of Mrs. Frey's calculator manufacturing company as a function of the year, we can use a linear equation, assuming that the sales increase at a constant rate each year. We have two data points: $4.2 million in 2012 and $8.7 million in 2015.
Let's assume the year is represented as
�
t, and the sales in millions of dollars is represented as
�
(
�
)
S(t). The equation for the sales as a function of the year would be:
�
(
�
)
=
�
�
+
�
S(t)=mt+b
Where:
�
m is the rate of increase in sales per year.
�
b is the initial value of sales in the base year.
Using the data points we have:
In 2012 (t = 0), the sales were $4.2 million, so:
�
(
0
)
=
�
(
0
)
+
�
=
�
=
4.2
S(0)=m(0)+b=b=4.2
In 2015 (t = 3), the sales were $8.7 million, so:
=
8.7
S(3)=m(3)+
3
�
=
8.7
−
4.2
3m=8.7−4.2
3
�
=
4.5
3m=4.5
�
=
4.5
3
=
1.5
m=
3
4.5
=1.5
So, the rate of increase in sales per year (
�
m) is 1.5 million dollars.
Now that we have the equation for sales as a function of the year:
�
(
�
)
=
1.5
�
+
4.2
S(t)=1.5t+4.2
To predict the sales for her company in 2020 (t = 8 years after 2012), you can plug in
�
=
8
t=8 into the equation:
�
(
8
)
=
1.5
(
8
)
+
4.2
=
12
+
4.2
=
16.2
million dollars
S(8)=1.5(8)+4.2=12+4.2=16.2 million dollars
So, according to the equation, the sales for her company in 2020 are predicted to be $16.2 million.To represent the sales of Mrs. Frey's calculator manufacturing company as a function of the year, we can use a linear equation, assuming that the sales increase at a constant rate each year. We have two data points: $4.2 million in 2012 and $8.7 million in 2015.
Let's assume the year is represented as
�
t, and the sales in millions of dollars is represented as
�
(
�
)
S(t). The equation for the sales as a function of the year would be:
�
(
�
)
=
�
�
+
�
S(t)=mt+b
Where:
�
m is the rate of increase in sales per year.
�
b is the initial value of sales in the base year.
Using the data points we have:
In 2012 (t = 0), the sales were $4.2 million, so:
�
(
0
)
=
�
(
0
)
+
�
=
�
=
4.2
S(0)=m(0)+b=b=4.2
In 2015 (t = 3), the sales were $8.7 million, so:
�
(
3
)
=
�
(
3
)
+
�
=
3
�
+
4.2
=
8.7
S(3)=m(3)+b=3m+4.2=8.7
Now, we can solve for
�
m:
3
�
=
8.7
−
4.2
3m=8.7−4.2
3
�
=
4.5
3m=4.5
�
=
4.5
3
=
1.5
m=
3
4.5
=1.5
So, the rate of increase in sales per year (
�
m) is 1.5 million dollars.
Now that we have the equation for sales as a function of the year:
�
(
�
)
=
1.5
�
+
4.2
S(t)=1.5t+4.2
To predict the sales for her company in 2020 (t = 8 years after 2012), you can plug in
�
=
8
t=8 into the equation:
�
(
8
)
=
1.5
(
8
)
+
4.2
=
12
+
4.2
=
16.2
million dollars
S(8)=1.5(8)+4.2=12+4.2=16.2 million dollars
So, according to the equation, the sales for her company in 2020 are predicted to be $16.2 million.