Tutor profile: David D.
Find the x-intercept of the line $$L_1$$, which is perpendicular to $$-4x + 3y = 12$$ and passes through the point $$(0,4)$$.
The first step of solving a problem like this is to determine the equation for the line $$L_1$$. The information we have is that $$L_1$$ is perpendicular to $$-4x + 3y = 12$$ and passes through the point $$(0,4)$$. Let's start by converting the line we're given into $$y=mx+b$$ format. $$-4x + 3y = 12$$ $$3y = 4x + 12$$ $$y = (4/3)x + 4$$ Knowing the equation for the line, we can determine the slope of $$L_1$$. Perpendicular lines have opposite, reciprocal slopes. The slope of the line we have is $$(4/3)$$. The opposite, reciprocal of $$(4/3)$$ is $$(-3/4)$$. Therefore, the slope of $$L_1$$ is $$(-3/4)$$. We now can say, $$L_1 = y = (-3/4)x + b$$ But, we still need to find the y-intercept, $$b$$. Luckily, we are given a point that $$L_1$$ passes through, so we can plug that into our equation to find $$b$$. $$y = (-3/4)x + b$$ $$4 = (-3/4)*(0) + b$$ $$b = 4$$ Now, we know our y-intercept. We also could have noticed that the point $$(0,4)$$ already lies on the y-axis, but it is always better to show our work. Our final equation for the line $$L_1$$ then becomes $$L_1 = y = (-3/4)x + 4$$ The problem asks for the x-intercept of this line. To find that, we can just plug $$0$$ in for $$y$$ and solve for $$x$$. $$0 = (-3/4)x + 4$$ $$ -4 = (-3/4)x$$, multiply both sides by (-4/3) $$x = 16/3$$ Our final answer is the x-intercept is $$16/3$$.
A man stands om the flat ground 100 m away from 173 m tall building. A plane is flying a ground distance of 900 m directly behind the building (1 km away from the man) at an altitude of 1.5 km. Can the man see the plane? (Do not take into account the height of the man)
The best way to solve this problem is to construct two triangles using the distances given in the question. The first triangle will be built from the man, the ground, and the building. The ground and the building meet at the base of the building forming a right angle. The height of the building is one leg of the triangle and the ground between the man and the building makes the other leg. The hypotenuse of this triangle is the line from the man to the top of the building. Therefore, we can begin to construct our triangle with the information we have. We know that the horizontal leg (the distance from the man to the building) is 100 m. We also know that the vertical leg (the height of the building) is 173 m. Using this information, we need to find the angle the hypotenuse makes with the ground. Let's call this angle $$theta$$. Using the triangle we have constructed, we can say that the tangent of $$theta$$ is the height of the building divided by the ground distance from the man to the building. This gives us, $$tan(theta) = 173/100$$ Using the inverse tangent function, we get $$theta = 60$$ deg (approximately) $$theta$$ represents the line of sight that the man has to the top of the building. Anything that is behind the building must have a line of sight angle greater than $$theta$$ or it will be blocked by the building. Let's check the man's line of sight angle to the plane. We know the plane is a horizontal distance of 1 km or 1,000 m away from the man and a vertical distance of 1.5 km or 1,500 m from the ground. Those two distances make up the legs of our triangle. Using the same process as with the building, we can calculate the line of sight angle from the man to the plane which we will call $$psi$$. After building our triangle, we can show that the tangent of $$psi$$ is the altitude of the plane divided by the horizontal distance between the man and the plane. This gives us, $$tan(psi) = 1500/1000$$ Using the inverse tangent function, we get $$psi = 56$$ deg (approximately) Since $$psi < theta$$, it can be concluded that the building blocks the line of sight from the man to the plane. Therefore, the answer to the question is no, the man cannot see the plane.
What is the simplified expression for the following? $$(3x^2 + 5x + 2)/(x^2 - 3x - 4)$$
The first step in solving a problem like this is checking that the numerator is not evenly divisible by the denominator. We can check this by first comparing the orders of the terms in the numerator and the denominator. We can see by looking at the numerator and the denominator that they both share an $$x^2$$ term, an $$x$$ term, and a ones term. Since the orders of these terms match, we do not have to worry about having an $$x$$ term in the quotient. Now, we can check if the numerator is a scalar multiple of the denominator, which we can do by comparing the coefficients of the like terms in the numerator and denominator. Comparing the coefficients of the $$x^2$$ term, we get $$3 / 1 = 3$$ Comparing the coefficients of the $$x$$ term, we get $$5 / (-3) = -5/3$$ or approximately -1.67 Comparing the coefficients of the ones term, we get $$ 2 / (-4) = -1/2$$ or 0.5 Since these ratios are not all the same, it can be concluded that the numerator is not evenly divisible by the denominator. Now, we can set our sights on simplifying the expression. Whenever we see polynomials like this, we should always be looking to factor. Our goal is to check if the numerator or denominator share any factors. If so they can be cancelled out to simplify the expression. We will use basic factoring techniques to simplify the numerator and the denominator. Since the denominator looks a little easier to factor, we will start there. $$x^2 - 3x - 4$$ Since the coefficient of the $$x^2$$ term is already 1, we can begin to fill in our new binomials. $$(x + a)(x + b)$$, where a and b are real numbers, can be positive or negative, and can be integers or fractions. We can make a list of the factors of the ones term, -4. This list would include 2, -2 1, -4 -2, 2 -4, 1 Since $$(x+a)(x+b) = (x+b)(x+a)$$ due to the Commutative Property of Multiplication for polynomials, we can eliminate duplicate pairs who simply have their order switched as those represent the same expression. That leaves 2, -2 1, -4 Now let's add these terms together to see if any of them sum to equal the coefficient of the $$x$$ term, -3. $$2 + (-2) = 0$$ $$1 + (-4) = -3$$ And we have a winner! The pair (1, -4) both multiplies to equal the ones term and sums to equal the coefficient of the $$x$$ term. So we can go ahead and plug those into the expression we'd built and check that it does indeed equal the denominator. $$(x + 1)(x - 4) = x^2 - 3x - 4$$ We can now rewrite our original expression as follows $$(3x^2 + 5x + 2) / ((x + 1)(x -4))$$ Now, instead of completely factoring the numerator, we can simply divide the numerator by each term in the denominator to see if the expression can be simplified any further. Just by looking at the ones term, the numerator is not divisible by the (x-4) term because -4 is not a factor of 2. However, the numerator could be evenly divisible by the (x+1) term because 1 is a factor of 2. Let's divide the numerator by the (x+1) term to see if it is a factor. Using long division of polynomials, we can show that $$(3x^2 + 5x + 2) / (x + 1) = 3x + 2$$ Written out in step form, that process goes as follows. $$(5x+2) / (x + 1) = 2$$ with a remainder of $$3x$$. Carrying that term with us, $$(3x^2 + 3x) / (x + 1) = 3x$$ with no remainder, so our answer is $$3x+2$$. That process shows us that the numerator can be written as $$(3x + 2)(x + 1)$$ Which multiplied out does in fact equal the original numerator, $$3x^2 + 5x + 2$$. Let's now rewrite our original equation with our factored out expressions. $$((3x+1)(x+1))/((x+1)(x-4))$$ We can see that the numerator and denominator both share an (x+1) term, so those can be cancelled out leaving us with $$(3x+1)/(x-4)$$ Using the same process as we used with the original equation, we can determine that this numerator is not evenly divisible by the denominator, so the expression can no longer be simplified. The final answer is: a simplified expression of $$(3x^2+5x+2)/(x^2-3x-4)$$ is $$(3x+2)/(x-4)$$
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