Math / Algebra

QuestionPoints: 0.5 of 1 Save The population of a certain country was approximately 25 million in 1900,150 million in 1950, and 300 million in 2000 . Construct a model for this data by finding a quadratic equation whose graph passes through the points $(0,25),(50,150)$ , and $(100,300)$ . Use this model to estimate the population in 2050.
Let $x$ be the number of years since 1900 and $y$ be the population in millions. Determine a quadratic equation whose graph passes through the points $(0,25),(50,150)$ , and $(100,300)$ $\square$ (Use integers or decimals for any numbers in the expression.)

Studdy Solution

STEP 1

1. The population data is modeled by a quadratic equation of the form $y = ax^2 + bx + c$ , where $x$ is the number of years since 1900 and $y$ is the population in millions.
2. The quadratic equation must pass through the points $(0,25)$ , $(50,150)$ , and $(100,300)$ .
3. We need to determine the coefficients $a$ , $b$ , and $c$ of the quadratic equation.

STEP 2

1. Set up the system of equations based on the given points.
2. Solve the system of equations to find the coefficients $a$ , $b$ , and $c$ .
3. Construct the quadratic equation using the coefficients found.
4. Use the quadratic equation to estimate the population in 2050.

STEP 3

Substitute the point $(0, 25)$ into the quadratic equation $y = ax^2 + bx + c$ .
$25 = a(0)^2 + b(0) + c$

STEP 4

Simplify the equation from STEP_1 to find $c$ .
$c = 25$

STEP 5

Substitute the point $(50, 150)$ into the quadratic equation $y = ax^2 + bx + c$ .
$150 = a(50)^2 + b(50) + 25$

STEP 6

Simplify the equation from STEP_3.
$150 = 2500a + 50b + 25$ $125 = 2500a + 50b$ $125 = 50(50a + b)$ $2.5 = 50a + b$

STEP 7

Substitute the point $(100, 300)$ into the quadratic equation $y = ax^2 + bx + c$ .
$300 = a(100)^2 + b(100) + 25$

STEP 8

Simplify the equation from STEP_5.
$300 = 10000a + 100b + 25$ $275 = 10000a + 100b$ $2.75 = 100a + b$

STEP 9

We now have a system of linear equations from STEP_4 and STEP_6:
1) $2.5 = 50a + b$ 2) $2.75 = 100a + b$
Subtract the first equation from the second to eliminate $b$ . $2.75 - 2.5 = 100a - 50a$ $0.25 = 50a$ $a = 0.005$

STEP 10

Substitute $a = 0.005$ back into the first equation $2.5 = 50a + b$ to find $b$ .
$2.5 = 50(0.005) + b$ $2.5 = 0.25 + b$ $b = 2.25$

STEP 11

Using the values of $a$ , $b$ , and $c$ found, construct the quadratic equation:
$y = 0.005x^2 + 2.25x + 25$

STEP 12

To estimate the population in 2050, substitute $x = 150$ (since 2050 is 150 years after 1900) into the quadratic equation.
$y = 0.005(150)^2 + 2.25(150) + 25$

STEP 13

Calculate the value of $y$ .
$y = 0.005(22500) + 2.25(150) + 25$ $y = 112.5 + 337.5 + 25$ $y = 475$
Solution: The estimated population in 2050 is $475$ million.

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Studdy Solution

STEP 1

STEP 2

1. Set up the system of equations based on the given points.
2. Solve the system of equations to find the coefficients $a$ , $b$ , and $c$ .
3. Construct the quadratic equation using the coefficients found.
4. Use the quadratic equation to estimate the population in 2050.

STEP 3

Substitute the point $(0, 25)$ into the quadratic equation $y = ax^2 + bx + c$ .
$25 = a(0)^2 + b(0) + c$

STEP 4

Simplify the equation from STEP_1 to find $c$ .
$c = 25$

STEP 5

Substitute the point $(50, 150)$ into the quadratic equation $y = ax^2 + bx + c$ .
$150 = a(50)^2 + b(50) + 25$

STEP 6

Simplify the equation from STEP_3.
$150 = 2500a + 50b + 25$ $125 = 2500a + 50b$ $125 = 50(50a + b)$ $2.5 = 50a + b$

STEP 7

Substitute the point $(100, 300)$ into the quadratic equation $y = ax^2 + bx + c$ .
$300 = a(100)^2 + b(100) + 25$

STEP 8

Simplify the equation from STEP_5.
$300 = 10000a + 100b + 25$ $275 = 10000a + 100b$ $2.75 = 100a + b$

STEP 9

We now have a system of linear equations from STEP_4 and STEP_6:
1) $2.5 = 50a + b$ 2) $2.75 = 100a + b$
Subtract the first equation from the second to eliminate $b$ . $2.75 - 2.5 = 100a - 50a$ $0.25 = 50a$ $a = 0.005$

STEP 10

Substitute $a = 0.005$ back into the first equation $2.5 = 50a + b$ to find $b$ .
$2.5 = 50(0.005) + b$ $2.5 = 0.25 + b$ $b = 2.25$

STEP 11

Using the values of $a$ , $b$ , and $c$ found, construct the quadratic equation:
$y = 0.005x^2 + 2.25x + 25$

STEP 12

To estimate the population in 2050, substitute $x = 150$ (since 2050 is 150 years after 1900) into the quadratic equation.
$y = 0.005(150)^2 + 2.25(150) + 25$

STEP 13

Calculate the value of $y$ .
$y = 0.005(22500) + 2.25(150) + 25$ $y = 112.5 + 337.5 + 25$ $y = 475$
Solution: The estimated population in 2050 is $475$ million.

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Studdy Solution

STEP 1

STEP 2

1. Set up the system of equations based on the given points.2. Solve the system of equations to find the coefficients aaa, bbb, and ccc.3. Construct the quadratic equation using the coefficients found.4. Use the quadratic equation to estimate the population in 2050.

STEP 3

Substitute the point (0,25)(0, 25)(0,25) into the quadratic equation y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c.25=a(0)2+b(0)+c 25 = a(0)^2 + b(0) + c 25=a(0)2+b(0)+c

STEP 4

Simplify the equation from STEP_1 to find ccc.c=25 c = 25 c=25

STEP 5

Substitute the point (50,150)(50, 150)(50,150) into the quadratic equation y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c.150=a(50)2+b(50)+25 150 = a(50)^2 + b(50) + 25 150=a(50)2+b(50)+25

STEP 6

Simplify the equation from STEP_3.150=2500a+50b+25 150 = 2500a + 50b + 25 150=2500a+50b+25 125=2500a+50b 125 = 2500a + 50b 125=2500a+50b 125=50(50a+b) 125 = 50(50a + b) 125=50(50a+b) 2.5=50a+b 2.5 = 50a + b 2.5=50a+b

STEP 7

Substitute the point (100,300)(100, 300)(100,300) into the quadratic equation y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c.300=a(100)2+b(100)+25 300 = a(100)^2 + b(100) + 25 300=a(100)2+b(100)+25

STEP 8

Simplify the equation from STEP_5.300=10000a+100b+25 300 = 10000a + 100b + 25 300=10000a+100b+25 275=10000a+100b 275 = 10000a + 100b 275=10000a+100b 2.75=100a+b 2.75 = 100a + b 2.75=100a+b

STEP 9

STEP 10

Substitute a=0.005a = 0.005a=0.005 back into the first equation 2.5=50a+b2.5 = 50a + b2.5=50a+b to find bbb.2.5=50(0.005)+b 2.5 = 50(0.005) + b 2.5=50(0.005)+b 2.5=0.25+b 2.5 = 0.25 + b 2.5=0.25+b b=2.25 b = 2.25 b=2.25

STEP 11

Using the values of aaa, bbb, and ccc found, construct the quadratic equation:y=0.005x2+2.25x+25 y = 0.005x^2 + 2.25x + 25 y=0.005x2+2.25x+25

STEP 12

To estimate the population in 2050, substitute x=150x = 150x=150 (since 2050 is 150 years after 1900) into the quadratic equation.y=0.005(150)2+2.25(150)+25 y = 0.005(150)^2 + 2.25(150) + 25 y=0.005(150)2+2.25(150)+25

STEP 13

Calculate the value of yyy.y=0.005(22500)+2.25(150)+25 y = 0.005(22500) + 2.25(150) + 25 y=0.005(22500)+2.25(150)+25 y=112.5+337.5+25 y = 112.5 + 337.5 + 25 y=112.5+337.5+25 y=475 y = 475 y=475Solution: The estimated population in 2050 is 475475475 million.

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1. Set up the system of equations based on the given points.
2. Solve the system of equations to find the coefficients $a$ , $b$ , and $c$ .
3. Construct the quadratic equation using the coefficients found.
4. Use the quadratic equation to estimate the population in 2050.

Substitute the point $(0, 25)$ into the quadratic equation $y = ax^2 + bx + c$ .
$25 = a(0)^2 + b(0) + c$

Simplify the equation from STEP_1 to find $c$ .
$c = 25$

Substitute the point $(50, 150)$ into the quadratic equation $y = ax^2 + bx + c$ .
$150 = a(50)^2 + b(50) + 25$

Simplify the equation from STEP_3.
$150 = 2500a + 50b + 25$ $125 = 2500a + 50b$ $125 = 50(50a + b)$ $2.5 = 50a + b$

Substitute the point $(100, 300)$ into the quadratic equation $y = ax^2 + bx + c$ .
$300 = a(100)^2 + b(100) + 25$

Simplify the equation from STEP_5.
$300 = 10000a + 100b + 25$ $275 = 10000a + 100b$ $2.75 = 100a + b$

Substitute $a = 0.005$ back into the first equation $2.5 = 50a + b$ to find $b$ .
$2.5 = 50(0.005) + b$ $2.5 = 0.25 + b$ $b = 2.25$

Using the values of $a$ , $b$ , and $c$ found, construct the quadratic equation:
$y = 0.005x^2 + 2.25x + 25$

To estimate the population in 2050, substitute $x = 150$ (since 2050 is 150 years after 1900) into the quadratic equation.
$y = 0.005(150)^2 + 2.25(150) + 25$

Calculate the value of $y$ .
$y = 0.005(22500) + 2.25(150) + 25$ $y = 112.5 + 337.5 + 25$ $y = 475$
Solution: The estimated population in 2050 is $475$ million.