The temperature in Gavins oven is a sinusoidal function of t

The temperature in Gavin\'s oven is a sinusoidal function of time. Gavin sets his oven so that it has a maximum temperature of 290°F and a minimum temperature of 230°. Once the temperature hits 290°, it takes 20 minutes before it is 290° again. Gavin\'s cake needs to be in the oven for 30 minutes at temperatures at or above 280°. He puts the cake into the oven when it is at 260° and rising. How long will Gavin need to leave the cake in the oven? (Round your answer to the nearest minute.)

Solution

Hi :)

The general sine function format is:

A*sin(w*t)

what you know here is the frequency the and amplitude of the oven\'s temperature

we know the temperature fluctuates between 230-290, with a median at 260, 260 is 30 off of both of them, so we will consider 200 to be our \"y=0\" and 30 to be the amplitude

we also know that w = 2*pi*F
F = frequency, which we know that the oven goes through a 20 minute cycle so it has frequency of 1/20 cycles per minute

so f = 1/20

we now have an equation of

T = 30*sin(2*pi/20 * t)
we will add 260 to this so that our temperature match up and make it a little less confusing

T = 30*sin(2*pi/20 * t) + 260

so we need to know when this oven is going to reach 280 degrees, so set the equation to 280

280 = 30*sin(2*pi/20 * t) + 260

2/3 = sin(2pi*t/20)

do some rearranging to solve for t
t = 20*asin(2/3) / 2*pi
t = 2.3228 minutes

so we know it is in the oven for 2.3228 minutes before it heats up to 280 degrees. This is the up part of the sin wave.

We also know that the wave will also be 260 degrees halfway through its cycle, which is 10 minutes in. since sine waves are symmetric, we know it will stop being 280 degrees 2.3228 minutes before this point

so the leading and trailing 2.3228 minutes can be removed so we get

10-2.3228*2 = 5.3544 minutes. in the \"up\" part of the sine wave cycle
we know that the last 10 minutes of the cycle will all be below 260 degrees, so that is no good,

so basically we get a 5.3544 minutes per cycle if we take this, so 30 / 5.3544 = 5.6 cycles

from that we see that we need at least 5 cycles, not a full 6, so this means that he will take the cake out partway through.

we need to find that point

so if it went through a full 5 cycles we know it took 100 minutes at least,

so in those 5 cycles we got a total of 26.772 minutes, we need 30-26.772=3.228 more minutes of cooking time

we know we need to wait 2.3228 minutes into the next cycle before it hits 280, then we need to leave it in for another 3.228 minutes or basically take it out after 5.551 minutes into the 6th cycle

so we know the first five took 20 minutes each, totaling 100 minutes, then we let it go 5.551 minutes into the 6th cycle,

so we get a total of 105.551=106 minutes.

I hope it helps :)