what-is-the-total-surface-area-of-a-cube-if-the-length-of-each-main-diagonal-is-9-beginequation-begi

Question

What is the total surface area of a cube if the length of each main diagonal is 9?
\begin{equation}
\begin{array}{l}{	ext { (A) } 81} \ {	ext { (B) } 162} \ {	ext { (C) } 27 \sqrt{3}} \ {	ext { (D) } 81 \sqrt{3}}\end{array}
\end{equation}

Lily An · Accepted Answer

Okay, So for this one, it says, What is the total surface area of a cube if the length of each main diagonal is nine. So we know that the total surface area, So let&#x27;s go and defined some terms. So surface area, it&#x27;s just the total of all the areas of the surfaces. Okay, and then we know that main diagonal is the longest diagonal distance that we could draw in a cube. So here&#x27;s the thing is that some people think that the main diagonal, if this is our cube, is going to be this right here. But that&#x27;s not the main diagonal of Accu because that&#x27;s the Dragon Office Square. And also that&#x27;s not the longest it&#x27;s actually gonna be from here to here, like that intents of them. Let&#x27;s go in and draw out our figure and right over here. So here is my cube. Okay on, then, if the main diagnose or which is this is nine really to find the length so that we can find the surface areas because for a cube, we know that the length times the length is the area of one surface, and then there&#x27;s six surfaces, so we would just do six l squared for that one. So then, from here, we need to draw some lines in order to get that. So if we draw this line right here, then we know that this a vertical line to this platform right here is going to form a ninety degree angle. That might not really look like it, but you know that if we were to isolate it out and draw it, we know that it is. So then this right here is just one length. So let&#x27;s call that. Oh, and this is also something that we can find because that is made out of another right triangle, which is this part right here. Let&#x27;s go and draw out that one as well. So for this one is going to be l and L, which means by the Pythagorean theorem or special right triangles. This one is going to be route Tio assuming a crime, this group to its well and then in order to find our l, we just have to do. But that green here and one more time so jaggery. And then it says that l squared plus route to el that will quantity squared is equal to nine squared. So then we know that l squared and then this square in that square cancels out plus two l squared is equal to nine squared. And then from there we know that three l squared is equal to eighty one. So from here, I&#x27;m going to fight by three on both sides. To get that l squared is equal to twenty seven. And instead of square rooting this I&#x27;m just going to keep this acid is because l squared, we know is actually the the surface area uh, one square, because a square looks like this, it&#x27;s got fell. And so if we do, I&#x27;ll Square that&#x27;s the area. But then we know that there&#x27;s six of those in a cube and so we&#x27;re going to take that twenty seven and multiply that by six. And when we do that, we&#x27;re going to get one hundred and sixty two. So then we know that bee is our answer. So for this one, the key is to draw a couple of lines and make a couple of right triangles and then be able to say that you should be able to use the Pythagorean theorem

Michelle Nguyen · Answer

here We have some here. Like Cube. So defined a surface area of a cube. That formula is six a squared. So where to set him equal to each other. So 8 64 equals six a squared Gonna divide both sides by six. So that gives us 1 44 equals a squared would take the square root of both sides. So a is equal to 12. So the edge of the cube is 12.

Melissa Stefan · Answer

were asked to find the lateral area and the surface area of a cube with length nine inches. I drew a picture to help myself visualize what I&#x27;m doing. Let&#x27;s start with first writing the formulas for the lateral area and the surface area. We know that the lateral area of a prism is equal to the perimeter of the base of the prism times its height. And we know the surface area is equal to the lateral area, plus the area of both bases. So L plus to be. Now, if we identify what we know, we know that the perimeter of our prism that is a cube, the perimeter is going to be four times the side, which is four times nine inches, 36 the height. Because it&#x27;s a cube, the height has the same length as all for the base sides, which is going to be nine. So our lateral area is going to be equal to 36 times nine, which is 324 inches squared. We can use this value to substitute for finding our surface area. You have a 324 plus to whatever the area of the basis, while the area of the base of a square that&#x27;s actually pretty easy, it&#x27;s just the side squared that&#x27;s going to be nine squared, which is 81. So we&#x27;re gonna multiply anyone by two because we have two sides and then we are going to simplify. This is going to be 324 plus 162 and our surface area total is going to be 486 square inches.

Lily An · Answer

can, this one says. What is the volume in cubic inches of acute, whose total surface area is two hundred and sixteen square inches? First of all, let&#x27;s think about the equation for the surface area of a cute for a service area. If it&#x27;s a queue, you know that all the six sides are equal, so we could just take one of the sides areas and multiply that by six. So area on one side time. Six. So then what&#x27;s the area of one side? We know that because it&#x27;s a cube, it&#x27;s actually just made up of six squares, which means it&#x27;s just gonna be length times length, because the length and the width should be the same. So we could just call that length times width length when we know it&#x27;s gonna be six l squared, and we know that the six all squared in this particular Kew is going to be two sixteen square inches. Co. Tells us that the total surface area there&#x27;s two sixteen, so then we just have to first of all, isolate our l here. If we were to do that, we just have to divide both sides by six Look, which becomes thirty six. So another L squared is equal to thirty six. So from here we can try to get help by square routing both sides because this square in the square, it should cancel out. We should get that l. A is equal to six. So then I must think about the volume of a Q. So we know that the volume is just going to length times, width, times, height because we know length, width and height are all the same because it&#x27;s a cute you could just you tell cute. So if we plug in that l that we found from our surface area into this equation, we should get six cute, and that should just be two sixteen. So we know actually that the volume and the surface area actually be the same for this one.

August Miller · Answer

So we&#x27;re talking about cubes. The cube has a length or what in the height and of course is they&#x27;re all the same length. So we&#x27;ll simply represent this value is s for this problem. And the surface area of a cube is equal to six times side length squared and that the agonal of cube is equal to the side length times radical three. So, essentially, we simply have a system of equations and we&#x27;re now trying to represent D in terms of a So let&#x27;s turn a into an equation that is represented by us. We can divide out the six. So now we have s squared, is equal to a over six and then we&#x27;ll take the square root of both sides. So that&#x27;s is now equal to a radical A over six. Now we can substitute this new value for s into the equation for the diagonal. So D is equal to radical a over six times radical three. And now we have expressed the diagonal of the cube with its surface area

Ankit Gupta · Answer

so no, the f would you be their self? It area will be under hood off X squared plus big square which is equals to extrude toe and we know nor the diagonal diagonal is under Rudolf X Squared plus R squared. So know the radius off different can be bitterness under Rudolf Ex quipped plus X squared plus x credible substitute the are square property. So therefore we will get X Route three. So therefore, using the magnet our surfaced idea will be ex script. That means it would be de by Route three.

Problem

$$x^{2}+y^{2}+4 x-2 y=-1$$ The equation of a cir…

Problem 23 Hard Difficulty

Answer

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