1)let k and w be two consecutive integers such that k<x<w. If log base 7 of 143 = x, find the value of k+w
2) if 7 and -1 are two of the solutiosn for x in the equation 2x^3 +kx^2 -44x+w=0, find the value of k+w
3) from an ordinary deck of 52 cards, two cards are selected at random (without replacement). find the probability that both cards were hearts. express your answer as a common fraction reduced to lowest terms.
1. if log7 143 = x
then 7^x = 143
but 7^2 = 49 and 7^3 = 343
so k=2 and w = 3
then k+w = 5
2. let f(x) = 2x^3 + kx^2 - 44x + w
if 7 is a solution then f(7) = 0
2(343) + 49k - 308 + w = 0
49k + w = -378
if -1 is a solution, then f(-1) = 0
2(-1) + k + 44 + w = 0
k + w = -42
subtract them:
48k = -336
k = -7
in k+w=-42, -7+w = -42 ---> w = -35
and k+w = -42
3) prob(2 hearts) = (13/52)(12/51) = 1/17
thank you!
1) To solve this problem, we need to find the values of k and w based on the given equation log base 7 of 143 = x.
First, we need to understand logarithms. A logarithm is the inverse of an exponentiation. In this case, the logarithm base 7 of 143 represents the exponent to which 7 must be raised to obtain 143.
Therefore, we can rewrite the equation as 7^x = 143.
To find the value of x, we need to determine the power to which 7 is raised to obtain 143. Since 7^2 = 49 and 7^3 = 343, we can conclude that x = 3.
Now we can determine the values of k and w. Since k and w are consecutive integers such that k < x < w, we know that k = 2 and w = 4.
Finally, to find the value of k+w, we simply add k and w, which gives us 2 + 4 = 6.
Therefore, the value of k+w is 6.
2) To find the value of k+w based on the given equation 2x^3 + kx^2 - 44x + w = 0, where 7 and -1 are solutions for x, we can apply Vieta's formulas.
Vieta's formulas state that if a polynomial equation of degree n has solutions x1, x2, ..., xn, then the sum of the solutions is equal to the opposite of the coefficient of x^(n-1) divided by the coefficient of x^n. Also, the product of the solutions is equal to the constant term divided by the coefficient of x^n.
In this case, we know that 7 and -1 are solutions for x. Therefore, the sum of the solutions is 7 + (-1) = 6.
Now let's expand the polynomial equation to match it with the general form of a cubic equation: 2x^3 + kx^2 - 44x + w = 0.
Comparing coefficients, we can conclude that the coefficient of x^2 is k and the constant term is w.
According to Vieta's formulas, the sum of the solutions (k+w) is equal to -k divided by 2. Since we know that the sum of the solutions is 6, we can write the equation as -k/2 = 6.
Solving for k, we multiply both sides of the equation by -2, which gives us k = -12.
To find the value of w, we simply substitute the value of k into the equation k+w=6. So, -12 + w = 6.
Solving for w, we add 12 to both sides of the equation, which gives us w = 18.
Therefore, the value of k+w is -12 + 18 = 6.
So, the value of k+w is 6.
3) To find the probability that both cards selected at random (without replacement) from a standard deck of 52 cards are hearts, we need to determine the number of favorable outcomes (hearts) and the total number of possible outcomes.
In a standard deck of 52 cards, there are 13 hearts. When the first card is selected, there are 52 cards in total and 13 hearts. After one heart is selected, there are 51 cards left, including 12 hearts.
The probability of selecting a heart on the first draw is 13/52 or 1/4.
The probability of selecting a heart on the second draw, without replacement, is 12/51.
To find the probability of both events occurring, we multiply the probabilities together: (1/4) * (12/51) = 12/204.
Simplifying the fraction, we get 1/17.
Therefore, the probability that both cards selected are hearts is 1/17.