If 3x+2y=18 find x and y for which the product xy takes its maximum value
1 answer:
Answer:
(x, y) = (3, 4.5)
Step-by-step explanation:
You want to find x and y such that xy is a maximum subject to the constraint that 3x+2y=18.
<h3>Objective function</h3>The objective function f(x, y) = xy is most easily maximized if it can be written in terms of one variable only. Solving the constraint equation for y, we have ...
3x +2y = 18
2y = 18 -3x
y = 1/2(18 -3x) = (3/2)(6 -x)
Then the objective function is ...
f(x) = x(3/2)(6 -x)
<h3>Maximum</h3>We recognize this as the equation of a parabola that opens downward, and has zeros at x=0 and x=6. Its line of symmetry is halfway between those, at x = 3. That is the location of the maximum of the objective function.
The corresponding y-value is ...
y = (3/2)(6 -3) = 9/2 = 4.5
The values of x and y that maximize their product are (x, y) = (3, 4.5).
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<em>Additional comment</em>
In the attached graph, the larger the product, the farther away the curve is from the origin. The product is maximized subject to the constraint when the curves are tangent to each other. That point of tangency is (x, y) = (3, 4.5).
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Answer:
The weight of an adult male who is 172 cm tall is 89.36 kg.
Step-by-step explanation:
The regression equation representing the relationship between height and weight of a person is:
Here
<em>y</em> = weight of a person
<em>x</em> = height of a person
The information provided is:
The hypothesis to test the significance of the correlation between height and weight is:
<em>H₀</em>: There is no relationship between the height and weight, i.e. <em>ρ</em> = 0.
<em>Hₐ</em>: There is a relationship between the height and weight, i.e. <em>ρ </em>≠ 0.
Decision rule:
If the <em>p</em>-value of the test is less than the significance level, then the null hypothesis will be rejected and vice-versa.
According to information provided:
<em>p</em>-value = 0.023 < <em>α</em> = 0.05
The null hypothesis was rejected at 5% level of significance.
Thus, concluding that there is a relationship between the height and weight.
Compute the weight of an adult male with height, <em>x</em> = 172 cm as follows:
Thus, the weight of an adult male who is 172 cm tall is 89.36 kg.