The boy wants to build rectangular beetle cages to study beetle behavior. He wants all the sides on the base of his cages to be
1 answer:
Answer:
32 cm by 18 cm
16 cm by 36 cm,
48 cm by 12 cm
24 cm by 24 cm
Step-by-step explanation:
<u><em>The complete question is</em></u>
A boy wants to build rectangular cages. He wants all the sides on the base of his cages to beat least 10 cm long.he also wants the base area to equal 576 cm2. The boy needs help to find all the possible lengths and widths for the base of the cages. He used only whole centimeters, with no fractional parts.what are all the possible dimensions for the rectangular base?
we know that
The number 576 decompose in prime factors is the same that write
To find out possible side lengths, multiply different combinations of these prime numbers together (make sure each pair adds up to 10)
3x3=9cm 2x2x2x2x2x2= 64cm ----> is not a combination (one side is not greater than 10)
3x3x2= 18cm 2x2x2x2x2= 32cm
3x3x2x2= 36cm 2x2x2x2= 16cm
3x2x2= 12cm 3x2x2x2x2= 48cm
3x2x2x2= 24cm 3x2x2x2= 24cm
therefore
32 cm by 18 cm
16 cm by 36 cm,
48 cm by 12 cm
24 cm by 24 cm
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Answer:
y = -x - 6
Step-by-step explanation:
If we are looking for the equation of the line tangent to the circle, we have to find the first derivative of the circle function. If that tangent line is to be parallel to the given tangent line, we need to find the slope of the the given tangent line and make sure the slope of the line we are looking for is the same. First let's find the slope of the given tangent. If the given tangent is x+y=6, then to find the slope, solve for y:
y = -x + 6. So the slope of that line, and also the slope of the tangent we are solving for, is -1. Hold that thought while we find the derivative of the function. Using implicit differentiation, we find the derivative to be:
Solving for the slope (dy/dx) gives us:
and
so
Subbing in the value of the slope we found:
so
-y = -x or, equivalently,
y = x. Now that we know that y and x are the same value, we can go back to the original circle to find out what they both are by substitution. If y = x, we make the substution:
If , then
and
and
so
y = ±3
We will choose the negative root (you'll see why in a second) and sub that in for both y and x since y and x are th same number. Going back to the fact that the slope is -1:
y - (-3) = -1(x - (-3)) simplifies a bit to:
y + 3 = -x - 3 which gives us, in slope-intercept form:
y = -x - 6
That is the equation of the tangent to the circle that is parallel to the given tangent.
If we would have chosen the principle (or positive) root of 3, our equation would have looked like this:
y - 3 = -1(x - 3) and
y = -x + 3 + 3 so
y = -x + 6. Notice that that is the EXACT SAME EQUATION as the given tangent. That's why we have to pick the negative 3 as our root.
Good luck in your calculus class! Make sure you post your questions in here! Many of us LOVE the challenge of calculus!