Let's begin with recalling that the perimeter of a rectangle is the doubled sum of its length and width, and the area of a rectangle is the product of its length and width. Let l be the length of the table and w be its width.
P= 2(l+w)
A=l w
A model car builder wants a table with a perimeter of 26 feet and an area of no more than 30 square feet. Using this information we can write an equation and an inequality.
2(l+w)= 26
l w≤ 30
Since we are asked to determine the possible width of this table, we can solve the above equation for l and substitute its value into the inequality.
Test a value from each interval to see if it satisfies the original inequality.
Step 1
We will start by solving the related equation.
13w-w^2=30 ⇔ - 1w^2+ 13w+( -30)=0
We see above that a= -1, b= 13, and c= -30. Let's substitute these values into the Quadratic Formula to solve the equation.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
w=-13±7/-2
w=-13 + 7/-2
w=-13 - 7/-2
w=-6/-2
w=-20/-2
w=3
w=10
Step 2
The solutions of the related equation are 3 and 10. Remember that since w and l are dimensions they cannot be non-positive numbers. Therefore w must be greater than 0 and less than 13. Having this in mind, let's plot the solutions on a number line. Since the original is a non-strict inequality, the points will be closed.
Step 3
Finally, we must test a value from each interval to see if it satisfies the original inequality. Let's choose a value from the first interval, 0< w ≤ 3. For simplicity, we will choose w=1.
