The 8-ft wall shown here stands 27 ft from the building. Find ...

Question

The 8-ft wall shown here stands 27 ft from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.

beam

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a ladder is leaning against a house reaching over a 10 ft tall fence that is 24 ft from the house. What is the minimum length of the ladder needed to reach the house over the fence? Round your answer to two decimal places. Below the problem we're given a diagram of what was described in the question. Now, in order to solve this problem, we need to define a couple of quantities on our diagram. So we're going to label the length of the ladder as L. We're going to label the vertical distance from the ground to the point where the ladder touches the house as H, and we're going to label the horizontal distance from the fence to where the ladder touches the ground as X. And so what we're going to do is write. A function for the length of a ladder L in terms of X and H, and we want to minimize that function. In order to minimize that function, we'll just need it to be a function of one variable. So we need to relate X and H together. And we can do that using similar triangles since we basically have two right triangles in our diagram. We have the right triangle formed between the ground, the ladder, and the fence, and the right triangle between the ground that's formed from the ground, the ladder, and the house. So, recall for similar triangles, that the ratio of the sides must be equal in the two triangles. So in our case here, we're going to have H divided by X plus 24. And that's gonna be equal to. 10 divided by X. And so we're going to solve this equation for H. So to do that, we'll multiply both sides of the equation by the quantity of X plus 24. Doing so, we'll have H as equal to 10 divided by X, multiplied by the quantity of X plus 24. And so find the expression, that means that H is going to be equal to 10 + 240 divided by X. So here we have a relationship now between H and X. We can also relate L2X and H using Pythagorean's theorem since L here is going to be the hypotenuse of a right triangle. So that means that L is going to be equal to the square roots. Of the quantity of H2 plus the quantity of X plus 24 in quantity squared substituting in our expression for H, that means that L is going to be equal to the square roots. Of the quantity of the quantity of 10 plus 240 divided by X in quantity squared plus the quantity of X plus 24 in quantity squared in quantity. So now we have L just as a single variable of X. And we want to minimize L with respect to X. And if we minimize the quantity that's underneath the radical, that means we're also going to minimize L. So we're just going to focus on the quantity that is inside the radical. So we're going to call that quantity F of X. And that's going to be equal to the quantity of 10 plus 240 divided by X in quantity squared, plus quantity of X plus 24 in quantity squared. And so we want to minimize F of X here with respect to X. So to do that, we're going to take a derivative. So that means we need to calculate F of X. And that's going to be equal to the derivative with respect to X. Of the quantity of the quantity of 10 plus 240 divided by X in quantity squared, plus the quantity of X plus 24 in quantity squared in quantity. So taking this derivative, this is going to be equal to. To take the derivative of the first one, we're going to need to use our power rule as well as our chain rule. So applying the power rule, we'll have 2 multiplied by the quantity of 10 plus 240 divided by X in quantity, and we'll need to apply our chain rule, so we need to multiply this by the derivative with respect to x of the quantity of 10 plus 240 divided by X in quantity. And when you take that derivative, that's going to be equal to minus 240 divided by X squared. Then we'll have plus. When we take the derivative of the second term, we're also going to need to apply our power rule as well as our chain rule. So applying the power rule, we're going to have 2 multiplied by the quantity of X plus 24 in quantity, and we'll need to apply our chain rule as well. So we'll take this, we'll need to multiply this by the derivative with respect to x of the quantity of X plus 24. And when we take that derivative, that's just going to be equal to 1. So, we'll just need to simplify this expression. So this is going to be equal to 2 multiplied by X plus 24 in quantity. Minus 480 divided by X2, multiplied by the quantity of 10 plus 240 divided by X in quantity. So now we want to do is find the critical points, because that will give us the possible locations of uh local extrema. And recall that critical points occur whenever our first derivative is equal to 0 or when it's undefined. Now, our first derivative is going to be undefined when X is equal to 0, since we'd have a 0 in the denominator of our fraction. However, X equal to 0 is outside the domain of a function F of X, since X needs to be a quantity that is actually greater than 0 to make physical sense. So since that falls outside the domain over function F of X, that means that we don't need to consider it as a critical point. So that means that the only critical point is going to be due to our first derivative being equal to 0. So we're going to set to multiply the quantity of X plus 24 in quantity minus 480 divided by x2 multiplied by the quantity of 10 plus 240 divided by X. In Quanti Equal to 0. And now we need to solve for X. So, the first thing that we can do is actually multiply both sides of the equation by X cubed and divide both sides by 2. In doing so, we will get X cubed. Multiplied by the quantity of X plus 24. -240. And here we're going to multiply the quantity inside the prints here by X, so we'll have the multiplied the 240 by The quantity of 10 x. Plus 240 in 20 equal to 0. And so now what we can do is factor out a 10 out of the second term. That's inside the parenthesis, so we'll have X cubed, multiplied by X plus 24. Then we'll have minus. 240 Multiplied by 10, multiplied by the quantity of X plus 24 in quantity, and that all has to be equal to 0. So now I can factor out a quantity of X plus 24 out of each of our terms. And doing so, I'll have the quantity of X plus 24. In quantity multiplied by the quantity of X cubed minus 2400 in quantity, and that has to be equal to 0. So now what we need to do is set each one of these factors equal to 0 and solve for x. So for the first term, we'll have X plus 24 equal to 0. That means that X is equal to -24, and that's not a valid solution since we can't have negative values for X. So we're not going to consider that as a solution. So the other option is that we have X cubed minus 2400. Equal to 0, and when we solve for X, we get X is equal to the cube root. Of 2400 So now we need to determine if this X value is a local minimum. And to do that, we just need to determine whether the sign changes for a function F of X, when X is equal to square root, the cube root of 2400. And the sign of FX is going to be determined by the quantity X cubed minus 2400. Since X + 24, which is going to be the other factor. is always going to be a positive quantity since X must be a positive value. And the only difference between these two factors and our original function is a factor of 2, which is again a positive quantity. So if you look at the case when X is less than the cube root of 2400, that means X cubed is going to be less than 2400, and so when I subtract 2400 from X cubed, that means that I'm going to get a negative value. So, since that's going to be negative, that means that F of X is going to be less than 0. And if we look at the case when X is greater than the cube root of 2400. Again, if we look at X cube minus 2400, when X is greater than the cube root of 2400, that means X cube is going to be greater than 2400, which means when I subtract 2400 from that quantity, I'm going to get a positive quantity. So that means that F of X is going to be greater than 0. And so here we see that our first derivative is changing signs at X equal to the cube root of 2400, and it's changing from negative to positive. So that means we do have a local minimum at X equal to the cube root of 2400. So now all that we need to do now that we've found the minimum value is to calculate our length L. And so that means that L is going to be equal to the square roots. Of the quantity Of the quantity of 10. Plus 240 divided by X, which in this case is going to be the cube root. Of 2400. In quantity squared plus The quantity of X, which is the cube root of 2400 plus 24 in quantity squared in quantity. And when we evaluate this expression, this is going to be approximately equal to 46.67, and this has units of feet since all of our distances were given in feet. So this here is the answer to the problem. So I hope that this explanation has been useful, and I'll see you in the next video.

The 8-ft wall shown here stands 27 ft from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.

Key Concepts

Pythagorean Theorem

Optimization

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