Tutor profile: Zach D.
Subject: Physics (Newtonian Mechanics)
A dog's displacement has components x and y expressed as functions of time t in the following equations: x(t) = 3t^2 - 4.5t + 9 y(t) = -t^2 + 5t - 4 What is the magnitude of the dog's acceleration after 3 seconds?
We can find equations for velocity and acceleration from the dog's displacement by finding the first and second derivatives of displacement, respectively. For velocity: dx/dt = vx(t) = 6t - 4.5 dy/dt = vy(t) = -2t + 5 Then for acceleration, the second derivative: dvx/dt = ax(t) = 6 dvy/dt = ay(t) = -2 We see that the acceleration of the dog is constant, irrespective of time. So the magnitude of its acceleration after 3 seconds is the same as its magnitude at any other given time. To find the magnitude, we simply use vector addition: magnitude of a = sqrt(6^2 + (-2)^2) = 2*sqrt(10)
Given the equation: y = 5x^-3 + 68 At what point is the slope of the tangent line perpendicular to the line given by equation: 2x + 5y = 7
Let's start with understanding which slope we are trying to find. We know it needs to be perpendicular to the line equation we are given. We can find the slope of our line equation by doing some algebra to get it from point-slope form to y-intercept form. 2x + 5y = 7 -2x -2x 5y = -2x + 7 y = (-2/5)x + (7/5) The slope of this line is -2/5. Recall that the slopes are perpendicular if they are negative reciprocals of each other. 5/2 is perpendicular to -2/5. Thus, we want our slope of the tangent line to equal 5/2. We will come back to this. First lets find an equation for the slope fo the tangent line, which is our derivative. y' = -15x^-4 . solved using the power rule (multiple exponent by coefficient and subtract exponent by 1) This equation represents the slope of the tangent line, we know it's value should equal to 5/2. 5/2 = -15x^-4 but our question asks for a point, let's solve for x 5x^4 = -30 from cross multiplying x = -6^(1/4) Now that we have the x value of our point, we can use the original equation to find the y value. y = 5(6)^(-3/4) + 68 y = (5/6)*6^(1/4) + 68 Our point is a bit messy, but it comes out to be: (-6^(1/4), (5/6)*6^(1/4) + 68)
Given the equations: 3x + 4y = 18 -5x - y = 2 Solve for both values of x and y.
This is a system of equations. One of the reasons people find this kind of problem difficult is there are multiple ways you can solve it. But no worries! That just means you have options. All options, however, involve you (1) picking an equation, (2) solving for a variable of that equation, (3) using substitution, and (4) solving for the other variable. Let's see what this looks like in action. (1) I'll pick -5x - y = 2 because the coefficients are smaller. (2) Let's choose to solve for y. I chose this because y's coefficient has an absolute value of 1. If I think about steps ahead, I won't have to divide by its coefficient to solve for it. -5x - y = 2 +y +y add y to both sides -5x = 2 + y -2 -2 subtract 2 on both sides -5x - 2 = y or y = -5x - 2 (3) Now that we know an expression that represents y, we can use the other equation: (3x + 4y = 18) that we haven't used yet. We can take the expression that we know y equals (-5x - 2) and substitute it into the equation 3x + 4y = 18 as follows: 3x + 4(-5x - 2) = 18 Something great happened during this step! We have only x in the equation so now we can solve for it! 3x -20x - 8 = 18 distributing the 4 -17x - 8 = 18 combine like terms +8 +8 add 8 on both sides -17x = 26 divide both sides by -17 to solve for x x = -26/17 -26/17 is not a fraction we can simplify so we can leave it as is. (4) x = -26/17 and we can plug this value into either of the original equations to solve for y. But let's wait a second. In step (2) we already found an expression for y so let's use that. y = -5x - 2 y = -5(-26/17) - 2 plug in the value that we know x equals y = -30/17 - 2 multiply the fraction by -5 130/17 cannot be simplified, 17 is a prime number! We need to turn 2 into a fraction with a denominator of 17 before we can combine the values. Remember 2 has a denominator of 1 so we can write it like this: 2/1 If we multiple top (numerator) and bottom (denominator) by 17 we will get: 34/17 y = 130/17 - 34/17 now we can combine our numerators y = 96/17 Our final solutions are: x = -26/17 y = 96/17 Great job!!! P.S. You can plug these values into both original equations to double check that they work :]
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