Recommended that temperature stay below 135F. Temperature (T) Minutes after begin operation (x) modeled by T = -0.005x^2 + 0.45x + 125

Question

Tell whether the temperature of the part will ever reach or exceed 135F. Use discriminant of a quadratic equation to decide. Part 2: If machine is in operation for 90 minutes before turning off, how many times will the temperature be 134F

 

 

Answer 1

Already answered.  No need to post twice.

Answer 2

a)

 

Set T=135

 

-0.005x² + 0.45x + 125 = 135

 

 

Subtract 135 on both sides of the equation.

 

-0.005x² + 0.45x - 10 = 0

 

 

The discriminant is        b² - 4ac

 

where:

a = -0.005

b = 0.45

c = -10

 

If the discriminant is negative, then the temperature will never reach 135 degrees.

If the discriminant is zero, then the temperature reaches or exceeds 135 degrees.

 

 

b)

 

 

Use the quadratic formula

 

where:

a = -0.005

b = 0.45

c = -9

 

Then compare the solutions.  If both real solutions fall in the interval 0<x≤90, then the temperature will be 134 degrees two times.  If only one solution falls in the interval, then only one time.

Answer 3

Hey Xavier.

 

Part 1) When it says to use the Discriminant, it wants you to take a look at the value of (b² - 4ac) when T is 135. So,

 

135 = -0.005x² + 0.45x + 125

0 = -0.005x² + 0.45x - 10

 

b² - 4ac = (0.45)² - 4*(-0.005)(-10)

 

= 0.2025 - 0.200 = 0.0025

 

Since this value is greater than or equal to zero, you know it will exist somewhere. If it were a negative result, then you'd know that this equation only has imaginary roots, and thus no real value of x will allow T to be 135.

 

Part 2) This is asking you to solve the equation:

 

134 = -0.005x² + 0.45x + 125

 

on the interval [0,90]. It doesn't care about any value beyond x = 90 minutes because the machine won't be operating past that time anyway.

 

So

 

0 = -0.005x² + 0.45x - 9

 

I'm going to divide everything by -0.005 to get rid of the ugly numbers.

 

0 = x² - 90x + 1800

 

-1800 = x² - 90x

 

Complete the square

 

-1800 + 2025 = x² - 90x + 2025

 

225 = (x - 45)²

 

+/- 15 = x - 45

 

x = 45 + 15

x = 45 - 15

 

x = 60, 30

 

The temp will reach 134 deg F twice in the 90 minute time span; once at 30 minutes, once at 60 minutes.

 

Hope this helps.