The UV index on a sunny day can be modeled by the function f(x)=-0.15(x-13)^2+7.6 where x represents the time of day on a 24-h clock and f(x) represents the UV index. At what time(s) was the UV index 7? (2 marks)
- 0.15 ( x - 13 ) ^ 2 + 7.6 = 7 Subtract 7.6 to both sides
- 0.15 ( x - 13 ) ^ 2 + 7.6 - 7.6 = 7 - 7.6
- 0.15 ( x - 13 ) ^ 2 = - 0.6 Divide both sides by - 0.15
( x - 13 ) ^ 2 = - 0.6 / - 0.15
( x - 13 ) ^ 2 = 4 Take square root to both sides
x - 13 = + OR - 2 Add 13 to both sides
x - 13 + 13 = + OR - 2 + 13
x = + OR - 2 + 13
The solutions are :
x = - 2 + 13 = 11
and
x = 2 + 13 = 15
To find the time(s) when the UV index is 7, we need to solve the equation f(x) = 7.
Given the function: f(x) = -0.15(x-13)^2 + 7.6
Setting f(x) = 7, we have:
7 = -0.15(x-13)^2 + 7.6
Subtracting 7 from both sides:
0 = -0.15(x-13)^2 + 7.6 - 7
Simplifying:
0 = -0.15(x-13)^2 + 0.6
To continue solving this equation, we'll isolate the quadratic term:
0.15(x-13)^2 = 0.6
Dividing both sides by 0.15:
(x-13)^2 = 4
Taking the square root of both sides:
x - 13 = ±2
Now, we can solve for x:
For x - 13 = 2:
x = 2 + 13
x = 15
For x - 13 = -2:
x = -2 + 13
x = 11
Thus, the UV index is 7 at two different times: 11:00 and 15:00.
To find the time(s) when the UV index is 7, we need to solve the equation f(x) = 7 using the given function f(x) = -0.15(x-13)^2+7.6.
Step 1: Set up the equation
We start by substituting f(x) with 7 in the equation:
-0.15(x-13)^2+7.6 = 7
Step 2: Simplify the equation
Next, we simplify the equation by expanding and rearranging terms:
-0.15(x^2 - 26x + 169) + 7.6 = 7
-0.15x^2 + 3.9x - 25.35 + 7.6 = 7
-0.15x^2 + 3.9x - 17.75 = 0
Step 3: Solve for x
We can now solve the quadratic equation -0.15x^2 + 3.9x - 17.75 = 0 using factoring, completing the square, or the quadratic formula.
Let's solve it using the quadratic formula:
The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -0.15, b = 3.9, and c = -17.75. We can substitute these values into the formula:
x = (-3.9 ± √(3.9^2 - 4(-0.15)(-17.75))) / (2(-0.15))
Simplifying further:
x = (-3.9 ± √(15.21 - 10.065)) / -0.3
x = (-3.9 ± √5.145) / -0.3
Using a calculator to evaluate √5.145 ≈ ±2.27, we have:
x ≈ (-3.9 - 2.27) / -0.3 ≈ 20.23
x ≈ (-3.9 + 2.27) / -0.3 ≈ 3.88
Step 4: Interpret the results
We have found two approximate solutions for x: 20.23 and 3.88.
Since x represents time on a 24-hour clock, the time(s) when the UV index is 7 are approximately:
1) Around 3:53 AM (3.88 on the 24-hour clock)
2) Around 8:14 PM (20.23 on the 24-hour clock)