Given that a and b are nonnegative real numbers such that a+2b=60, what is the largest possible value of ab?
And
If f(x) = x^2 - 3x + 4, find the value of
f(sqrt3 - sqrt2).
And
The median of {20, x, 15, 30, 25} is 0.4 less than the mean. If x is a whole
number, what is the sum of all possible values of x?
If a+2b=60 that is just like the problem of finding the maximum area enclosed on three sides by 60 ft of fencing. That is when a=30 and 2b=30. So, maximum ab = 30*15 = 450
f(x) = x^2 - 3x + 4
f(√3-√2) = (√3-√2)^2 - 3(√3-√2) + 4
The mean is (90+x)/5 = 18 + 0.2x
arranged in order (15,20,x,25,30)
x = 18+0.2x
x = 22.5
But x can only be one of 20,21,22,23,24,25
Only x=23 satisfies median=mean-0.4
the mean of (15,20,23,25,30) = 22.6
I still don't understand the second one, can you explain it more please?
I've gotten to 1-sqrt 3+sqrt 2
but I don't know what to do next, or if it's wrong.
(√3-√2)^2 = 3-2√6+2 = 5-2√6
so that means you have
5-2√6 - 3√3 + 3√2) + 4
= 9 + 3√2 - 3√3 - 2√6
Oh, that makes a lot more sense thank you
To find the largest possible value of ab, we need to use the constraint a+2b=60. Let's solve this equation for a in terms of b:
a = 60 - 2b
Now, we can rewrite ab in terms of b only:
ab = (60 - 2b)b = 60b - 2b^2
To find the largest possible value of ab, we need to find the maximum value of this quadratic function. We can do this by finding the vertex of the parabola formed by the function. The vertex is given by the formula:
x = -b / 2a
In our case, a = -2 and b = 60, so substituting these values into the formula:
x = -60 / (2*(-2)) = -60 / -4 = 15
The largest possible value of ab occurs when b = 15. Substituting this value into the expression for ab, we get:
ab = 60(15) - 2(15^2) = 900 - 450 = 450
Therefore, the largest possible value of ab is 450.
For the second question, we are given f(x) = x^2 - 3x + 4 and we need to find the value of f(sqrt3 - sqrt2).
To find this value, we will substitute sqrt3 - sqrt2 into the function f(x) and simplify.
f(x) = (sqrt3 - sqrt2)^2 - 3(sqrt3 - sqrt2) + 4
Expanding the square and distributing the terms:
= (3 - 2sqrt3sqrt2 + 2) - 3sqrt3 + 3sqrt2 + 4
= 9 - 6sqrt6 + 2 - 3sqrt3 + 3sqrt2 + 4
= 15 - 6sqrt6 - 3sqrt3 + 3sqrt2
Therefore, f(sqrt3 - sqrt2) = 15 - 6sqrt6 - 3sqrt3 + 3sqrt2.
For the third question, we are given a set {20, x, 15, 30, 25} and we know that the median of this set is 0.4 less than the mean.
The median is the middle value of a set, which means that after arranging the numbers in ascending order, if the set has an odd number of elements, the median is the middle element. If the set has an even number of elements, the median is the average of the two middle elements.
Arranging the set in ascending order, we get {15, 20, 25, 30, x}. Since 15 and 30 are at the extremes, the median will be the middle element, which is 25.
The mean of a set is found by summing all the elements and dividing by the number of elements. In our case, the mean is (15 + 20 + 25 + 30 + x) / 5.
Given that the median is 0.4 less than the mean, we can form the equation:
25 = (15 + 20 + 25 + 30 + x) / 5 - 0.4
Multiplying both sides by 5:
125 = 15 + 20 + 25 + 30 + x - 2
Combining like terms:
x = 125 - 15 - 20 - 25 - 30 + 2
x = 37
Therefore, the sum of all possible values of x is 37.