+2446 You must understand the trigonometric functions to solve this problem. We can use these because we know that we have a right angle because it is given in the diagram:
| \(\frac{\tan 56}{1}=\frac{x}{26}\) | The tangent function compares the opposite angle to the adjacent one. Cross multiply to isolate x. |
| \(26\tan 56=x\) | Use a calculator to approximate the distance to the tenth, as instructed. |
| \(x\approx38.5ft\) | Of course, don't forget your units on your final answer! |
You are done! YEAH. I also feel as if I have answered this exact question before; I guess I am glad that I am answering it again.
+2446 I think I have an explanation for you that may help you. I have a picture that should help you visualize this phenomenon
Source: https://vt-s3-files.s3.amazonaws.com/uploads/problem_question_image/image/1471/square_diagonal.jpg
A square is defined as an equiangular and equilateral quadrilateral. This is complicated language. Let's break this down, so it is easier to understand.
"Equiangular" means that all angle measures are congruent and have the same measure. Of course, in a square the measure of all the angles is \(90^{\circ}\), so a square satisfies this condition
"Equilateral" means that all sides are congruent and have the same length. This is also true in a square.
"Quadrilateral" means a four-sided figure. This fits the description of a square.
In the diagram above, let's find the length of \(d\), the length of the diagonal of a square when the side lengths have length \(a\). We'll use Pythagorean's Theorem for the length. \(\angle B\) is a right angle, so the hypotenuse is d, and the legs are a. Let's do it:
| \(a^2+a^2=d^2\) | Both side lengths are equivalent in a square. Let's solve for d by combining both a^2. |
| \(2a^2=d^2\) | Take the square root of both sides to isolate d. |
| \(\sqrt{2a^2}=d\) | |
Ok, the distance of diagonal of a square is \(\sqrt{2a^2}\). We can actually simplify this further. Let's do that:
| \(\sqrt{2a^2}\) | First, I'm going to use a radical rule stating \(\sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}\hspace{3mm},a\geq0\hspace{1mm},b\geq0\). Note that this radical rule can only be used if and only if both a and b are nonnegative. Therefore, this radical rule is not universal. However, we can assume that a is nonegative because of the context of geometry; it is nonsensical for a to have a side length of -8, for example. |
| \(\sqrt{2}\sqrt{a^2}\) | I'll use another radical rule stating that \(\sqrt[n]{a^n}\hspace{3mm},a\geq0\). This, too, can only be used for nonnegative values. Note that this rule make the \(\sqrt{a^2}\) turn into something much prettier. |
| \(a\sqrt{2}\) | |
Therefore, \(d=a\sqrt{2}\). There is something very special about this relationship. Do you know what this is? Let's look at 45-45-90 triangles:
Source: http://d2r5da613aq50s.cloudfront.net/wp-content/uploads/372023.image1.jpg
Do you notice any similiarities to this diagram and the one above? I do! The side lengths are the same and the length of d, the diagonal of the square and the hypotenuse of the triangle is equal to \(a\sqrt{2}\)! The converse of the Pythagorean theorem says that if \(a^2+b^2=c^2\), then \(\triangle ABC\) is right. The converse of the 45-45-90 triangle says if the side lengths of a triangle are equal and the length of the diagonal is that length times the square root of two, then the remaining angles are 45 degrees,
The proof is done now.
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-TheXSquaredFactor
+2446 I will simplify this expression, \(2(3x-1)-5(3x-1)\).
| \(2(3x-1)-5(3x-1)\) | This is the original expression. First, distribute the 2 into both terms in the parentheses. |
| \(6x-2-5(3x-1)\) | Distribute -5 into both terms in the other parentheses |
| \(6x-2-15x+5\) | Combine like terms. Combine the linear terms (the ones with an x) and the constants separately. |
| \(-9x+3\) | You are done! |
+2446 Changing all dimensions in area has a generalized effect; affecting the length and width by a scale factor of \(a\), the area will be multiplied by \(a^2\). Let's apply this idea:
| \(3.5^2=12.25\) | If the length and the width are being multiplied by 3.5, then the area is multiplied by that scale factor squared. |
| \(72*12.25\) | Multiply the original area by the scale factor squared to get the new area |
| \(882in^2\) | This is the new area of the rectagle. Of course, include units in your final answer. |
+2446 I will assume for this problem that i is for the imaginary number. If I should assume otherwise, tell me. I will, of course, simplify the given expression:
| \(e^{i*2*\pi}*(-1)^{98}*343^{\frac{1}{3}}\) | We will use an imaginary number rule here stating that \(e^{ia\pi}=(-1)^a, \text{so}\hspace{1mm}e^{i2\pi}=(-1)^2=1\). Of course, something multiplied by 1 is itself, so we are left with the other part. |
| \((-1)^{98}*343^{\frac{1}{3}}\) | -1 raised to an even power is always one, so this is another part of the multiplication that we can eliminate. |
| \(343^{\frac{1}{3}}\) | \(a^{\frac{1}{3}}=\sqrt[3]{a}\),so let's apply this rule, too. |
| \(\sqrt[3]{343}=7\) | 7 is your answer. |
