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kherson [118]
3 years ago
8

The hypotenuse of a right triangle is 37 units long. Find the other two sides if the perimeter of the triangle is 84 units. Sepa

rate the numbers by a comma.
Mathematics
1 answer:
taurus [48]3 years ago
7 0

Answer:

The lengths of the legs are 12 and 35 units

Step-by-step explanation:

Let c represent the hypotenuse and a and b represent the legs.  Then the Pythagorean Theorem requires that a^2 + b^2 = c^2 = 37^2 = Hypotenuse^2

Also:  a + b + c = Perimeter = 84 units.  

Since c = 37 units, a + b + 37 units = 84 units, or a + b = 47 units, or a = 47 - b.

Then a^2 + b^2 = 37^2 becomes (47 - b)^2 + b^2 = 37^2, or 1369.  Therefore:

2209 - 94b + b^2 + b^2 = 1369.

Simplifying by combining like terms, we get:  

840 - 94b + 2b^2 = 0, which is a quadratic equation in standard form.

Reducing all terms by division by 2, we get:

420 - 47b + b^2 = 0.  Here the coefficients are a = 1, b = -47 and c = 420.

The discriminant is therefore b^2 - 4ac, or 529, whose square root is ± 23.

Then the b values of this quadratic in b are

      47 ± 23

b = -------------- , so that b is either 35 or 12

             2

and then side a has length a = 47 - b = 12  or  35.

Thus, a = 12 and b = 35, and c is given:  37.

Check:  Is 12^2 + 35^2 = 37^2 true?  Is 144 + 1225 = 1369?  YES

The lengths of the legs are 12 and 35 units.

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Question: What equation models the data?
For this question I used a software to calculate the equation. When plotted, we can notice that the points form a parabolic function therefore we needed quadratic regression for this one. The quadratic equation is in the form of y=A+Bx+Cx^{2} where y represents the gas prices, x is the year, and A, B, and C are coefficients. I have found A, B, and C using the software and the equation for the data is:

y=-0.092+0.587x-0.026x^{2}

Question: What are the domain and range of the equation?
To find the range of the equation, we just transform the quadratic equation into the form y=a(x-h)^{2}+k. The value for k would be the maximum element in our range, and for the sake of the problem, let's assume that we don't have negative prices.

y=-0.092+0.587x-0.026x^{2}
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y=-0.026(x-11.289)^{2}+3.221

We can see that 3.221 is the maximum value of the range while zero is the minimum. Thus, we can express the range as {y|0 \leq y \leq 3.221}

For the domain, the number of years can extend infinitely but it will also depend if we want to consider the years where the price will be negative. Thus, let's just find the year when the price would be zero.

0=-0.026(x-11.289)^{2}+3.221
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Anywhere before and after the range of the x values above would result to a negative price therefore we can restrict the domain to those values. However, since x represents the year, we would need to round up the first value and round down the larger value.

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Question: Do you think your equation is a good fit for the data?
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Question: Does there seem to be a positive correlation, a negative correlation, or neither?
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