First you want to make an equation to express the problem. For this one in particular, let's make the length be represented by "L" and the width "W."
The length minus three times the width is 1 foot. This can be represented as L - 3W = 1. This means that L = 3W + 1. This will be our substitute for the length.
The perimeter of a rectangle is 2 times the length plus the width P = 2(L + W). By substituting these variables from what we now know we can find out the dimensions of the rectangle.
42 = 2(3W + 1 + W)
42 = 6W + 2 + 2W Multiply <u>everything in parenthesis</u> by 2.
42 = 8W + 2 Add the like terms.
40 = 8W Subtract 2 on both sides to get the <em>variable (W) </em>by itself.
5 = W Divide 8 from 40 to get what W equals.
The width is equal to 5. We can now plug in this answer to the original equation (P = 2(L + W)) or the length equation we made earlier (L = 3W + 1) to find the length.
42 = 2(L + 5)
42 = 2L + 10 Multiply <u>everything in parenthesis<em /></u><u><em /></u> by 2.<u />
32 = 2L Subtract 10 from both sides to get the <em>variable (L) </em>by itself.
16 = L Divide 2 from 32 to get what L equals.
<u>Or</u><u />
L = 3(5) + 1 Replace the <em>variable (W) </em>with 5.
L = 15 + 1 Multiply 3 by 5 to get 15.
L = 16 Add the remaining terms!
So the dimensions of the rectangle are 5 and 16, where 5 is the width and 16 is the length. I hope this answer made sense and that you can solve these on your own in the future!
4(x-10) .....................
By the law of sines, m∠<em>EFG</em> is such that
sin(m∠<em>EFG</em>) / (8 in.) = sin(m∠<em>G</em>) / (7.5 in)
so you need to find m∠<em>G</em>.
The interior angles to any triangle sum to 180°, so
m∠<em>DEG</em> = m∠<em>D</em> + m∠<em>G</em> + 43°
m∠<em>DEG</em> + m∠<em>D</em> + m∠<em>G </em>= 2 (m∠<em>D</em> + m∠<em>G</em>) + 43°
180° = 2 (m∠<em>D</em> + m∠<em>G</em>) + 43°
137° = 2 (m∠<em>D</em> + m∠<em>G</em>)
68.5° = m∠<em>D</em> + m∠<em>G</em>
But ∆<em>DEG</em> is isosceles, so m∠<em>D</em> = m∠<em>G</em>, which means
68.5° = 2 m∠<em>G</em>
34.25° = m∠<em>G</em>
<em />
Then
sin(m∠<em>EFG</em>) = (8 in.) sin(34.25°) / (7.5 in)
m∠<em>EFG</em> ≈ sin⁻¹(0.600325) ≈ 36.8932°
The angle is 162 degrees. Work shown in picture:)
