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Math  /  Algebra

QuestionUse the following information to answer the next question The power rating of a particular dynamic electronic circuit is given by the equation P=1−w−0.246tP=1-w^{-0.246 t} where PP is the power rating, tt is the amount of time since the circuit is switched on and ww is a constant.
2. After the circuit has been operational for 43 seconds, the power rating is 0.83 . The value of ww, to the nearest hundredth, is A. 0.09 B. 0.25 C. 1.18 D. 10.58

Studdy Solution

STEP 1

1. We are given the equation P=1−w−0.246t P = 1 - w^{-0.246t} .
2. The power rating P P after 43 seconds is 0.83.
3. We need to solve for the constant w w and round it to the nearest hundredth.

STEP 2

1. Substitute the given values into the equation.
2. Isolate the term involving w w .
3. Solve for w w .
4. Round the solution to the nearest hundredth.

STEP 3

Substitute P=0.83 P = 0.83 and t=43 t = 43 into the equation:
0.83=1−w−0.246×43 0.83 = 1 - w^{-0.246 \times 43}

STEP 4

Rearrange the equation to isolate the term involving w w :
0.83=1−w−10.578 0.83 = 1 - w^{-10.578}
Subtract 1 from both sides:
0.83−1=−w−10.578 0.83 - 1 = -w^{-10.578}
−0.17=−w−10.578 -0.17 = -w^{-10.578}
Multiply both sides by -1:
0.17=w−10.578 0.17 = w^{-10.578}

STEP 5

To solve for w w , take the reciprocal of both sides to eliminate the negative exponent:
w10.578=10.17 w^{10.578} = \frac{1}{0.17}
Calculate the reciprocal:
w10.578=5.882 w^{10.578} = 5.882
Take the 10.578 10.578 -th root of both sides to solve for w w :
w=5.882110.578 w = 5.882^{\frac{1}{10.578}}
Calculate the value:
w≈1.18 w \approx 1.18

STEP 6

Round w w to the nearest hundredth:
The value of w w is approximately 1.18 1.18 .
The correct answer is:
1.18 \boxed{1.18}

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