In a study during the

1
9
7
0
'
s
,
it was found that one model of legislative turnover in the country A parliament was described by the following formula. Upper M left parenthesis t right paren

Question

In a study during the
​
 
1
9
7
0
'
s
,
 ﻿it was found that one model of legislative turnover in the country A parliament was described by the following formula. Upper M left parenthesis t right parenthesis equals 
4
5
0
 ﻿e Superscript negative 
0
.
0
7
 ﻿tM
(
t
)
=
4
5
0
e
−
0
.
0
7
t In this formula
​
 ﻿M
(
t
)
 ﻿was the number of continuously serving members at time t
.
​
 ﻿Here, t equals 
0
t
=
0
 ﻿represents
​
 
1
9
6
5
,
 ﻿t equals 
1
t
=
1
 ﻿represents
​
 
1
9
6
6
,
 ﻿and so on
.
 ﻿Use this model to approximate the number of continuously serving members in each year. 
​
(
a
)
1
9
7
0
1
9
7
0
 
​
(
b
)
1
9
7
2
1
9
7
2
 
​
(
c
)
1
9
7
8
1
9
7
8
 ﻿Question content area bottom Part 
1
 
​
(
a
)
 ﻿Find the number of continuously serving members in 
1
9
7
0
1
9
7
0
.
 ﻿enter your response here 
​
(
Round to the nearest whole number as
​
 ﻿needed.
)
 ﻿Part 
2
 
​
(
b
)
 ﻿Find the number of continuously serving members in 
1
9
7
2
1
9
7
2
.
 ﻿enter your response here 
​
(
Round to the nearest whole number as
​
 ﻿needed.
)
 ﻿Part 
3
 
​
(
c
)
 ﻿Find the number of continuously serving members in 
1
9
7
8
1
9
7
8
.
 ﻿enter your response here 
​
(
Round to the nearest whole number as
​

VIPSolutions · Accepted Answer

Given:

- Model: $ M(t) = 450e^{-0.07t} $
- $ t = 0 $ is **1965**  
- $ t = 1 $ is **1966**, and so on.

Let's compute $ t $ for each year:

- (a) **1970**: $ t = 1970 - 1965 = 5 $
- (b) **1972**: $ t = 1972 - 1965 = 7 $
- (c) **1978**: $ t = 1978 - 1965 = 13 $

Now, plug these values into the formula, rounding to the nearest whole number.

---

### **(a) 1970 ($ t = 5 $):**

$$
M(5) = 450e^{-0.07 \times 5} = 450e^{-0.35}
$$

Calculate $ e^{-0.35} $:

$$
e^{-0.35} \approx 0.7047
$$

$$
M(5) \approx 450 \times 0.7047 \approx 317.1
$$

**Rounded:**  
$$
\boxed{317}
$$

---

### **(b) 1972 ($ t = 7 $):**

$$
M(7) = 450e^{-0.07 \times 7} = 450e^{-0.49}
$$

Calculate $ e^{-0.49} $:

$$
e^{-0.49} \approx 0.6120
$$

$$
M(7) \approx 450 \times 0.6120 \approx 275.4
$$

**Rounded:**  
$$
\boxed{275}
$$

---

$$
M(13) = 450e^{-0.07 \times 13} = 450e^{-0.91}
$$

Calculate $ e^{-0.91} $:

$$
e^{-0.91} \approx 0.4010
$$

$$
M(13) \approx 450 \times 0.4010 \approx 180.5
$$

**Rounded:**  
$$
\boxed{181}
$$

---

## **Summary Table**

| Year | $ t $ | $ M(t) $ (rounded) |
|------|--------|----------------------|
| 1970 | 5      | 317                  |
| 1972 | 7      | 275                  |
| 1978 | 13     | 181                  |

---

## **Final Answers**

- **(a) 1970:** **317**
- **(b) 1972:** **275**
- **(c) 1978:** **181**

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