or the period 1990-2003, the amount of biscuits, pastas, and noodles y (in thousands of metric tons)imported into the united states ca be modeled by the function y=1.36(xsquared)+ 27.8x+304 where x is the number of year since 1990.
a.write and solve an equation that you can use to approximate the year in which 500 million pounds of biscuits, pasta, and noodles were imported.
b. write and solve an equation that you can use to approximate the year in which 575 million pounds of biscuits, pasta and noodles were imported.
a. To convert the given function to pounds, we need to multiply the result by 2204.62 (since 1 metric ton = 2204.623 pounds).
The given function in pounds:
y = 1.36(2204.62)x^2 + 27.8(2204.62)x + 304(2204.62)
Now we want to find the year when 500 million pounds were imported:
500,000,000 = 2998.323x^2 + 61328.156x + 670000.48
Now, we can solve for x using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
where, a = 2998.323, b = 61328.156, and c = 670000.48 - 500,000,000
x = (-61328.156 ± √(61328.156^2 - 4(2998.323)(-499329999.52)))/(2*2998.323)
x ≈ 4.624 and x ≈ -45.909
Since the year cannot be negative, we take x = 4.624, which means that approximately in the year 1994 (1990 + 4.624), 500 million pounds of biscuits, pasta, and noodles were imported.
b. Now we will find the year when 575 million pounds were imported.
575,000,000 = 2998.323x^2 + 61328.156x + 670000.48
Now, we can solve for x using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
where, a = 2998.323, b = 61328.156, and c = 670000.48 - 575,000,000
x = (-61328.156 ± √(61328.156^2 - 4(2998.323)(-574329999.52)))/(2*2998.323)
x ≈ 6.514 and x ≈ -50.122
Since the year cannot be negative, we take x = 6.514, which means that approximately in the year 1996 (1990+6.514), 575 million pounds of biscuits, pasta, and noodles were imported.
a. To approximate the year in which 500 million pounds of biscuits, pasta, and noodles were imported, we need to set the equation equal to 500 million:
1.36x^2 + 27.8x + 304 = 500
Now, we solve for x by rearranging the equation and using quadratic formula:
1.36x^2 + 27.8x + 304 - 500 = 0
1.36x^2 + 27.8x - 196 = 0
Using the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a)
x = (-27.8 ± √(27.8^2 - 4 * 1.36 * -196))/(2 * 1.36)
After solving for x, you will obtain two solutions - one positive and one negative. Since we are interested in the number of years since 1990, only consider the positive solution.
b. To approximate the year in which 575 million pounds of biscuits, pasta, and noodles were imported, we follow the same process as in part a. The equation becomes:
1.36x^2 + 27.8x + 304 = 575
Again, solve for x using the quadratic formula:
1.36x^2 + 27.8x + 304 - 575 = 0
1.36x^2 + 27.8x - 271 = 0
Using the quadratic formula: x = (-b ± √(b^2 - 4ac))/(2a)
x = (-27.8 ± √(27.8^2 - 4 * 1.36 * -271))/(2 * 1.36)
Similarly, solve for x and consider only the positive solution for the number of years since 1990.
To approximate the year in which 500 million pounds of biscuits, pasta, and noodles were imported, we need to set up and solve the equation using the given function.
a. Let's start by setting up the equation:
y = 1.36x^2 + 27.8x + 304
Since y represents the amount of biscuits, pasta, and noodles imported in thousands of metric tons, and we want to find the year when 500 million pounds were imported, we need to convert pounds to thousands of metric tons.
1 million pounds = 1000 thousand pounds
500 million pounds = 500,000 thousand pounds
Therefore, the equation becomes:
500,000 = 1.36x^2 + 27.8x + 304
Now we can solve this equation to find the value of x, which represents the number of years since 1990.
To solve the quadratic equation, we can rearrange it to bring all the terms to one side:
1.36x^2 + 27.8x + 304 - 500,000 = 0
Next, we can simplify the equation:
1.36x^2 + 27.8x - 499,696 = 0
Now, we can solve this quadratic equation using various methods such as factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1.36, b = 27.8, and c = -499,696.
Plugging the values into the formula, we get:
x = (-27.8 ± √(27.8^2 - 4 * 1.36 * -499,696)) / (2 * 1.36)
Simplifying further:
x = (-27.8 ± √(772.84 + 2719644.16)) / 2.72
x = (-27.8 ± √(2720417)) / 2.72
Now, calculate the square root:
x = (-27.8 ± 1650.47) / 2.72
Finally, use the two resulting values of x to find the years since 1990.
b. The process for finding the year in which 575 million pounds of biscuits, pasta, and noodles were imported is the same as in part (a). Just replace the value of 500,000 with 575,000 in the equation:
575,000 = 1.36x^2 + 27.8x + 304
Then follow the same steps to solve for x and find the years since 1990.