<span>Maximum area = sqrt(3)/8
Let's first express the width of the triangle as a function of it's height.
If you draw an equilateral triangle, then a rectangle using one of the triangles edges as the base, you'll see that there's 4 regions created. They are the rectangle, a smaller equilateral triangle above the rectangle, and 2 right triangles with one leg being the height of the rectangle and the other 2 angles being 30 and 60 degrees. Let's call the short leg of that triangle b. And that makes the width of the rectangle equal to 1 minus twice b. So we have
w = 1 - 2b
b = h/sqrt(3)
So
w = 1 - 2*h/sqrt(3)
The area of the rectangle is
A = hw
A = h(1 - 2*h/sqrt(3))
A = h*1 - h*2*h/sqrt(3)
A = h - 2h^2/sqrt(3)
We now have a quadratic equation where A = -2/sqrt(3), b = 1, and c=0.
We can solve the problem by using a bit of calculus and calculating the first derivative, then solving for 0. But since this is a simple quadratic, we could also take advantage that a parabola is symmetrical and that the maximum value will be the midpoint between it's roots. So let's use the quadratic formula and solve it that way. The 2 roots are 0, and 1.5/sqrt(3).
The midpoint is
(0 + 1.5/sqrt(3))/2 = 1.5/sqrt(3) / 2 = 0.75/sqrt(3)
So the desired height is 0.75/sqrt(3).
Now let's calculate the width:
w = 1 - 2*h/sqrt(3)
w = 1 - 2* 0.75/sqrt(3) /sqrt(3)
w = 1 - 2* 0.75/3
w = 1 - 1.5/3
w = 1 - 0.5
w = 0.5
The area is
A = hw
A = 0.75/sqrt(3) * 0.5
A = 0.375/sqrt(3)
Now as I said earlier, we could use the first derivative. Let's do that as well and see what happens.
A = h - 2h^2/sqrt(3)
A' = 1h^0 - 4h/sqrt(3)
A' = 1 - 4h/sqrt(3)
Now solve for 0.
A' = 1 - 4h/sqrt(3)
0 = 1 - 4h/sqrt(3)
4h/sqrt(3) = 1
4h = sqrt(3)
h = sqrt(3)/4
w = 1 - 2*(sqrt(3)/4)/sqrt(3)
w = 1 - 2/4
w = 1 -1/2
w = 1/2
A = wh
A = 1/2 * sqrt(3)/4
A = sqrt(3)/8
And the other method got us 0.375/sqrt(3). Are they the same? Let's see.
0.375/sqrt(3)
Multiply top and bottom by sqrt(3)
0.375*sqrt(3)/3
Multiply top and bottom by 8
3*sqrt(3)/24
Divide top and bottom by 3
sqrt(3)/8
Yep, they're the same.
And since sqrt(3)/8 looks so much nicer than 0.375/sqrt(3), let's use that as the answer.</span>
Geometric sequence general form: a * r^n
For Greg, we are given the elimination of the medicine as a geometric nth term equation:
200 * (0.88)^t
The amount of medicine starts at 200 mg and every hour, decreases by 12%;
To compare the decrease in medicine in the body between the two, it is useful to get them in a common form;
So, using the data provided for Henry, we will also attempt to find a geometric nth term equation that will work if we can:
As a geometric sequence, the first term would be a and the second term would be ar where r = multiplier;
If we divide the second term by the first term, we will therefore get r, which is 0.94 for Henry;
We can check that the data for Henry can be represented as a geometric sequence by using the multiplier (r) to see if we can generate the third value of the data;
We do this like so:
282 * (0.94)^2 = 249.18 (correct to 2 d.p)
We can tell that the data for Henry is also a geometric sequence.
So now, we just look at the multiplier for Henry and we find that every hour, the medicine decreases by 6%, half of the rate of decrease for Greg.
The answer is therefore that <span>Henry's body eliminated the antibiotic at half of the rate at which Greg's body eliminated the antibiotic.</span>
Answer: 3x-2y=24 in slope intercept form is 
The graph of the equation is in the attachment
Step-by-step explanation: Slope intercept form is y = mx + b
m is the slope b is the y-intercept x is the "input" in the equation y is the "output" or the y-value in the graph for a given x-value.
You need to rearrange the standard form to slope-intercept to isolate y on the left side of the equation.
3x - 2y = 24 Standard form. Subtract 3x from both sides.
-2y = -3x + 24 divide both sides by -2
-2y/-2 = -3x/-2 + 24/-2 . Simplify.
y = 3x/2 - 12 .
To clarify the slope as a fraction
Answer: 1 minute and 43 seconds faster
Step-by-step explanation:
I think x=24 but not sure
