Math / Algebra

QuestionThe table shows projections for the female population of a country (in millions). \begin{tabular}{l|c|c|c} Year & 2020 & 2045 & 2050 \\ \hline Female Population & 181 & 227 & 236 \end{tabular} (a) Find a quadratic function $\mathrm{f}(\mathrm{x})=\mathrm{ax}{ }^{2}+\mathrm{bx}+\mathrm{c}$ that gives the female population (in millions) in year x , where $\mathrm{x}=0$ corresponds to the year 2000 . (b) Estimate the female population in the year 2030 . (a) The coefficient of $x^{2}$ in the equation is $a=$ $\square$ (Do not round until the final answer. Then round to three decimal places as needed.)

Studdy Solution

STEP 1

1. The problem involves finding a quadratic function $f(x) = ax^2 + bx + c$ that models the female population of a country over time.
2. The variable $x$ represents the number of years since 2000, so $x=0$ corresponds to the year 2000.
3. The given data points are: (20, 181), (45, 227), and (50, 236), where the first number in each pair is $x$ and the second number is the population in millions.

STEP 2

1. Set up a system of equations using the given data points and the quadratic function $f(x) = ax^2 + bx + c$ .
2. Solve the system of linear equations to find the coefficients $a$ , $b$ , and $c$ .
3. Form the quadratic function $f(x)$ using the values of $a$ , $b$ , and $c$ .
4. Use the quadratic function to estimate the female population in the year 2030 ( $x=30$ ).

STEP 3

Using the data point (20, 181), we set up the first equation by substituting $x = 20$ and $f(x) = 181$ into the quadratic function:
$181 = a(20)^2 + b(20) + c$

STEP 4

Simplify the equation from STEP_1:
$181 = 400a + 20b + c$

STEP 5

Using the data point (45, 227), we set up the second equation by substituting $x = 45$ and $f(x) = 227$ into the quadratic function:
$227 = a(45)^2 + b(45) + c$

STEP 6

Simplify the equation from STEP_3:
$227 = 2025a + 45b + c$

STEP 7

Using the data point (50, 236), we set up the third equation by substituting $x = 50$ and $f(x) = 236$ into the quadratic function:
$236 = a(50)^2 + b(50) + c$

STEP 8

Simplify the equation from STEP_5:
$236 = 2500a + 50b + c$

STEP 9

We now have a system of three linear equations:
1. $400a + 20b + c = 181$
2. $2025a + 45b + c = 227$
3. $2500a + 50b + c = 236$
Solve the system for $a$ , $b$ , and $c$ . First, subtract the first equation from the second to eliminate $c$ :
$2025a + 45b + c - (400a + 20b + c) = 227 - 181$

STEP 10

Simplify the equation from STEP_7:
$1625a + 25b = 46$

STEP 11

Next, subtract the second equation from the third to eliminate $c$ :
$2500a + 50b + c - (2025a + 45b + c) = 236 - 227$

STEP 12

Simplify the equation from STEP_9:
$475a + 5b = 9$

STEP 13

Now, solve the simplified system of two equations:
1. $1625a + 25b = 46$
2. $475a + 5b = 9$
First, solve the second equation for $b$ :
$475a + 5b = 9 \implies 5b = 9 - 475a \implies b = \frac{9 - 475a}{5}$

STEP 14

Substitute $b = \frac{9 - 475a}{5}$ into the first equation:
$1625a + 25\left(\frac{9 - 475a}{5}\right) = 46$

STEP 15

Simplify and solve for $a$ :
$1625a + 5(9 - 475a) = 46 \implies 1625a + 45 - 2375a = 46 \implies -750a = 1 \implies a = -\frac{1}{750}$

STEP 16

Using $a = -\frac{1}{750}$ , solve for $b$ :
$b = \frac{9 - 475\left(-\frac{1}{750}\right)}{5} = \frac{9 + \frac{475}{750}}{5} = \frac{9 + \frac{19}{30}}{5} = \frac{\frac{270 + 19}{30}}{5} = \frac{289}{150} = 1.9267$

STEP 17

Substitute $a$ and $b$ into any of the original equations to solve for $c$ . Using $400a + 20b + c = 181$ :
$400\left(-\frac{1}{750}\right) + 20 \left(1.9267\right) + c = 181$

STEP 18

Simplify and solve for $c$ :
$- \frac{400}{750} + 38.534 + c = 181 \implies - \frac{8}{15} + 38.534 + c = 181 \implies c = 181 - 38.534 + \frac{8}{15} \implies c = 142.466 + \frac{8}{15} \implies c = 142.466 + 0.53333 = 143$

STEP 19

The quadratic function is:
$f(x) = -\frac{1}{750}x^2 + 1.9267x + 143$

STEP 20

To estimate the female population in the year 2030 ( $x=30$ ), substitute $x = 30$ into the quadratic function:
$f(30) = -\frac{1}{750}(30)^2 + 1.9267(30) + 143$

STEP 21

Simplify the expression:
$f(30) = -\frac{1}{750}(900) + 57.801 + 143 \implies f(30) = -1.2 + 57.801 + 143 \implies f(30) = 199.601$
Solution: (a) The coefficient of $x^2$ in the equation is $a = -\frac{1}{750} \approx -0.001$ (to three decimal places). (b) The estimated female population in the year 2030 is approximately $199.601$ million.

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Studdy Solution

STEP 1

STEP 2

STEP 3

Using the data point (20, 181), we set up the first equation by substituting x=20x = 20x=20 and f(x)=181f(x) = 181f(x)=181 into the quadratic function:181=a(20)2+b(20)+c 181 = a(20)^2 + b(20) + c 181=a(20)2+b(20)+c

STEP 4

Simplify the equation from STEP_1:181=400a+20b+c 181 = 400a + 20b + c 181=400a+20b+c

STEP 5

Using the data point (45, 227), we set up the second equation by substituting x=45x = 45x=45 and f(x)=227f(x) = 227f(x)=227 into the quadratic function:227=a(45)2+b(45)+c 227 = a(45)^2 + b(45) + c 227=a(45)2+b(45)+c

STEP 6

Simplify the equation from STEP_3:227=2025a+45b+c 227 = 2025a + 45b + c 227=2025a+45b+c

STEP 7

Using the data point (50, 236), we set up the third equation by substituting x=50x = 50x=50 and f(x)=236f(x) = 236f(x)=236 into the quadratic function:236=a(50)2+b(50)+c 236 = a(50)^2 + b(50) + c 236=a(50)2+b(50)+c

STEP 8

Simplify the equation from STEP_5:236=2500a+50b+c 236 = 2500a + 50b + c 236=2500a+50b+c

STEP 9

STEP 10

Simplify the equation from STEP_7:1625a+25b=46 1625a + 25b = 46 1625a+25b=46

STEP 11

Next, subtract the second equation from the third to eliminate ccc:2500a+50b+c−(2025a+45b+c)=236−227 2500a + 50b + c - (2025a + 45b + c) = 236 - 227 2500a+50b+c−(2025a+45b+c)=236−227

STEP 12

Simplify the equation from STEP_9:475a+5b=9 475a + 5b = 9 475a+5b=9

STEP 13

STEP 14

Substitute b=9−475a5b = \frac{9 - 475a}{5}b=59−475a​ into the first equation:1625a+25(9−475a5)=46 1625a + 25\left(\frac{9 - 475a}{5}\right) = 46 1625a+25(59−475a​)=46

STEP 15

Simplify and solve for aaa:1625a+5(9−475a)=46 ⟹ 1625a+45−2375a=46 ⟹ −750a=1 ⟹ a=−1750 1625a + 5(9 - 475a) = 46 \implies 1625a + 45 - 2375a = 46 \implies -750a = 1 \implies a = -\frac{1}{750} 1625a+5(9−475a)=46⟹1625a+45−2375a=46⟹−750a=1⟹a=−7501​

STEP 16

STEP 17

Substitute aaa and bbb into any of the original equations to solve for ccc. Using 400a+20b+c=181400a + 20b + c = 181400a+20b+c=181:400(−1750)+20(1.9267)+c=181 400\left(-\frac{1}{750}\right) + 20 \left(1.9267\right) + c = 181 400(−7501​)+20(1.9267)+c=181

STEP 18

STEP 19

The quadratic function is:f(x)=−1750x2+1.9267x+143 f(x) = -\frac{1}{750}x^2 + 1.9267x + 143 f(x)=−7501​x2+1.9267x+143

STEP 20

To estimate the female population in the year 2030 (x=30x=30x=30), substitute x=30x = 30x=30 into the quadratic function:f(30)=−1750(30)2+1.9267(30)+143 f(30) = -\frac{1}{750}(30)^2 + 1.9267(30) + 143 f(30)=−7501​(30)2+1.9267(30)+143

STEP 21

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Using the data point (20, 181), we set up the first equation by substituting $x = 20$ and $f(x) = 181$ into the quadratic function:
$181 = a(20)^2 + b(20) + c$

Simplify the equation from STEP_1:
$181 = 400a + 20b + c$

Using the data point (45, 227), we set up the second equation by substituting $x = 45$ and $f(x) = 227$ into the quadratic function:
$227 = a(45)^2 + b(45) + c$

Simplify the equation from STEP_3:
$227 = 2025a + 45b + c$

Using the data point (50, 236), we set up the third equation by substituting $x = 50$ and $f(x) = 236$ into the quadratic function:
$236 = a(50)^2 + b(50) + c$

Simplify the equation from STEP_5:
$236 = 2500a + 50b + c$

Simplify the equation from STEP_7:
$1625a + 25b = 46$

Next, subtract the second equation from the third to eliminate $c$ :
$2500a + 50b + c - (2025a + 45b + c) = 236 - 227$

Simplify the equation from STEP_9:
$475a + 5b = 9$

Substitute $b = \frac{9 - 475a}{5}$ into the first equation:
$1625a + 25\left(\frac{9 - 475a}{5}\right) = 46$

Simplify and solve for $a$ :
$1625a + 5(9 - 475a) = 46 \implies 1625a + 45 - 2375a = 46 \implies -750a = 1 \implies a = -\frac{1}{750}$

Substitute $a$ and $b$ into any of the original equations to solve for $c$ . Using $400a + 20b + c = 181$ :
$400\left(-\frac{1}{750}\right) + 20 \left(1.9267\right) + c = 181$

The quadratic function is:
$f(x) = -\frac{1}{750}x^2 + 1.9267x + 143$

To estimate the female population in the year 2030 ( $x=30$ ), substitute $x = 30$ into the quadratic function:
$f(30) = -\frac{1}{750}(30)^2 + 1.9267(30) + 143$