Highway engineers often use quadratic functions to model safe stopping distances for cars like 1/2x quantity squared + 11/5x is used to model the safe stopping distance for car traveling at x miles per hour on dry, level pavement. If a driver can see only 200 feet ahead on a highway with a sharp curve, then safe drivers speed x satisfy the quad inequality 1/2x quantity squared + 11/5x is less than or equal to 200. Solve this inquality to determine safe stopping speeds in a curve where a driver can see the road ahead at most 200 ft. what might be a safe speed limit for this curve?

Question

Highway engineers often use quadratic functions to model safe stopping distances for cars like 1/2x quantity squared + 11/5x is used to model the safe stopping distance for car traveling at x miles per hour on dry, level pavement. If a driver can see only 200 feet ahead on a highway with a sharp curve, then safe drivers speed x satisfy the quad inequality 1/2x quantity squared + 11/5x is less than or equal to 200. Solve this inquality to determine safe stopping speeds in a curve where a driver can see the road ahead at most 200 ft. what might be a safe speed limit for this curve?

Answer 1

I cannot determine exactly your first term.

Put the quadratic in this form

ax^2+bx<=200
then
ax^2+bx-200<=0
then solve the left side

x=(-b+-sqrt(b^2-4ac)/2a

Answer 2

To solve the quadratic inequality and determine the safe stopping speeds in a curve where a driver can see the road ahead at most 200 ft, we can follow these steps:

Step 1: Write the inequality equation:
1/2x^2 + 11/5x ≤ 200

Step 2: Multiply through by 10 to clear the fraction:
5x^2 + 22x ≤ 2000

Step 3: Rearrange the equation in standard form and set it equal to zero:
5x^2 + 22x - 2000 ≤ 0

Step 4: Solve the quadratic equation:
Since the inequality is less than or equal to zero, we need to find the values of x that make the quadratic expression negative or zero.

To solve the equation, you can either factor it, complete the square, or use the quadratic formula. In this case, factoring may not be straightforward, so we will use the quadratic formula.

Using the quadratic formula: x = (-b ± √(b^2 - 4ac))/2a, where in our equation, a = 5, b = 22, and c = -2000.

x = (-22 ± √(22^2 - 4 * 5 * -2000))/(2 * 5)
x = (-22 ± √(484 + 40000))/10
x = (-22 ± √40484)/10

From here, we can simplify further to get the two roots:
x = (-22 + √40484)/10 and x = (-22 - √40484)/10

Step 5: Solve for the safe stopping speeds:
Since the driver cannot have a negative speed, we only need to consider the positive root. So, we have:

x = (-22 + √40484)/10

Calculating the positive root using a calculator, we find:
x ≈ 24.78 mph

Therefore, a safe speed limit for this curve may be around 24.78 mph to ensure that the driver can stop within 200 ft given the quadratic model.