Please, I'm confuse! Farmer Ted has an isosceles right triangular garden. He is planning to increase the size of his garden by making one leg 7 feet longer and the other leg 5 feet longer. The area of Ted’s new garden will be 55 square feet larger than the original garden. How long were the original legs of the garden?
Answers
OK. You asked for it. Here we go.
First, let's gather up the tools we might use ...
some things that we know about triangles:
-- Every triangle: Area = (1/2) x (length of the base) x (the height)
-- Isosceles triangle: It has two sides that are the same length.
-- Right triangle: It has one right angle in it.
The sides that meet at the right angle are called the "legs".
They form a corner there, like this _| .
-------------------------------------------
Now we can start using these tools to hack away at the problem.
Farmer Ted has an isosceles right triangle garden.
The problem asks us to figure out how long the legs are.
Before he changes anything, it looks like this _| and both of
those sides are the same length. Call it 'x' until we figure out
what it really is.
Notice that one of them is the base of the triangle, and the other one
is the height. So the area of this triangle is
(1/2) (x) (x) or (1/2) x² .
-------------------------------
Farmer Ted is never satisfied. Suddenly, one day without warning, he
comes along and makes the garden bigger. He makes one of the legs
7 ft longer, and he makes the other one 5 ft longer.
Now the length of one leg is (x + 7) and the other one is (x + 5) .
They're still the base and height of the triangle, so the area of the
bigger garden is
(1/2) (x + 7) (x + 5).
The problem says that this area is 55 square feet more than the original
area, so look out, here comes the equation :
new area = old area + bigger
(1/2) (x + 7) (x + 5) = (1/2) x² + 55
Locked in the mysterious shadowy crevices of this equation is
everything we need in order to figure out the original length of the legs ...
what we called 'x'.
At this point, we can forget about Farmer Ted, forget about the garden,
and just go back to our laboratory with this equation and solve it to find 'x'.
Let's take it slow and easy, one little step at a time:
(1/2) (x + 7) (x + 5) = (1/2) x² + 55
Multiply each side by 2 : (x + 7) (x + 5) = x² + 110
Expand (FOIL) the left side: x² + 12x + 35 = x² + 110
Subtract x² from each side: 12x + 35 = 110
Subtract 35 from each side: 12x = 75
Divide each side by 12 : x = 6.25 feet
==============================================
OK. That's a very nice number. How do we know whether it's correct ?
Let's check it out, and see if it fits the story:
Original area = (1/2 x base x height) = (0.5 x 6.25 x 6.25) = 19.53125 sq ft
One new leg = (6.25 + 7) = 13.25 ft
Other new leg = (6.25 + 5) = 11.25 ft
New area = (1/2 x base x height) = (0.5 x 13.25 x 11.25) = 74.53125 sq ft
How much bigger is the new area ?
74.53125 - 19.53125 = 55 sq ft yay !
When we start with legs that are 6.25-ft and go through the whole story,
the new area is exactly what the problem says it was. So 6.25-ft is the
correct original length of the legs, before Farmer Ted messed with it.
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x(4x-3x⁴+1)
After that, although it's not required, it's best to put the terms in order of their power:
x(-3x⁴+4x+1)
x(-3x⁴+4x+1) is your final answer.
Answers
2.686
14/16
Answers
The fraction 14/16 in simplest form is 7/8.
To simplify the fraction 14/16, find the greatest common divisor (GCD) of the numerator (14) and the denominator (16) and then divide both the numerator and denominator by the GCD.
Step 1: Find the Greatest Common Divisor (GCD) of 14 and 16.
The divisors of 14 are 1, 2, 7, and 14.
The divisors of 16 are 1, 2, 4, 8, and 16.
The largest number that appears in both lists is 2. Therefore, the GCD of 14 and 16 is 2.
Step 2: Divide both the numerator and denominator by the GCD (which is 2).
14 ÷ 2 = 7
16 ÷ 2 = 8
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To find a fraction's simplest form, you need to find the GCF. The greatest common factor is the largest number that can go into the numerator and the denominator.
We know that 1 is a factor, but let's keep going to make sure. 2 is also a factor.
14/2 = 7
16/2 = 8
Since 14 and 16 are not divisible by 3, 2 is the GCF (we know not to continue dividing).
Therefore, it's simplest form is 7/8.
Answers
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50 - 20 = 3r - r
30 = 2r
Divide by 2 to isolate r and to find its value
2 and 2 cancels out
15 = r
Check, if you want :)
50 + (15) = 3(15) + 20
65 = 45 + 20
65 = 65