Step-by-step solutions to hundreds of high school geometry questions

Recent questions in High school geometry

Chaz Blair 2022-05-21 Answered

Given a point on a hyperbolic paraboloid, prove that there exist exactly 2 lines that pass through that point and lie on the surface of that paraboloid.

Kendrick Pierce 2022-05-21 Answered

You can use the dot product to determine if two vectors are parallel by seeing if their dot product is = 1. What about if they are parallel but point in the opposite direction? Would the dot product just equal -1 in this case?

Aditya Erickson 2022-05-21 Answered

I got a plane with equation: $P (x, y) = \frac{16100 + 47 y}{52}$
A line (parametrized equation) which lies on the plane P:
${\begin{matrix} x = t \\ y = 620 \\ z = 870 \end{matrix}$
And a point which belongs to the line: $O (- 5, 620, 870)$
I need to find an equation of the line that belongs to the specified plane P and crosses specified line at the specified point O making an angle of $45^{o}$ with the line.

Ethan Lee 2022-05-20

Find the surface area of the prism.

Jhendy Mae Estrada 2022-05-19

indimiamimactjcf 2022-05-15 Answered

I need to prove the maxima of the following summation, using Lagrange.
$max_{x_{m}} (\sum_{m} a_{m} l o g (x_{m}))$
s.t.
$0 \leq x_{m} \leq 1$
$\sum_{m} x_{m} = 1$
The solution is a closed form $x_{m} = \frac{a_{m}}{\sum_{m} a_{m}}$ .

I formulated the Lagrange equation but I am confused about the signs and the multipliers.

$L (x, λ, μ) = \sum_{m} a_{m} l o g (x_{m}) + \sum_{m} λ_{m} (1 - x_{m}) + μ (\sum_{m} x_{m} - 1)$ , Is this formulation correct ? what is wrong ?

note: only one $μ$ for one constraint.

Micah Haynes 2022-05-14 Answered

Converting between the $y = e + d x$ and $a x + b y = c$ forms of a line equation

Osmarq5ltp 2022-05-14 Answered

I have a non-linear maximization problem and I want to convert it to be a minimization problem, can I do so by multiplying it by a negative sign, or is that wrong; and if that is wrong what should I do?

arbixerwoxottdrp1l 2022-05-14 Answered

Plane with normal vector
a) If unit normal vector is $(a_{1}, b_{1}, c_{1}),$ , then, how the point $P_{1}$ on the plane becomes $(D a_{1}, D b_{1}, D c_{1}) ?$ ?
b) If unit normal is $(1 / 3, 2 / 3, 2 / 3)$ then $P_{1}$ becomes $(2 / 3, 4 / 3, 4 / 3)$ Where $D = 2.$ We know that normal vector began on the plane at point $P_{1}$ and ends at $(1 / 3, 2 / 3, 2 / 3) .$ My questions is how unit normal vector coordinates value less than $P_{1}$ coordinates value, because unit normal vector pointing outside of the plane it should be greater coordinates value than $P_{1}$ ?

2022-05-13

2022-05-12

What is the volume of this cone?

Use 𝜋 ≈ 3.14 and round your answer to the nearest hundredth.

2 ft
4 ft

cubic feet

Noelle Wright 2022-05-10 Answered

We have a function:
$f (x, y, z) = x \cos (ω t + y + ϕ_{1}) + z \cos (ω t + y + ϕ_{2}) .$
I want to solve the following maximization problems:
$max_{x, y, z} max_{t} f (x, y, z) or max_{x, y, z} \int_{< T >} f (x, y, z) d t .$
Surely, $x$ and $z$ are nonnegative, and $y$ is in $[0, 2 π)$ .

Can someone give me hints for solving the problem, or let me know some references to solve this types of problem?

jistefaftexia99kq6 2022-05-10 Answered

Let $f (x, y) : R \times R \mapsto R$ be a continuous and differentiable function. When can we claim that the following holds true:
$\frac{d}{d x} max_{y \in R} f (x, y) = max_{y \in R} \frac{\partial}{\partial x} f (x, y)$
assuming that both $max_{y \in R} f (x, y)$ and $max_{y \in R} \frac{\partial}{\partial x} f (x, y)$ exists.

Osmarq5ltp 2022-05-10 Answered

Each cow weighs 300kg. The cost of maintaining a cow is 20€ per day. The rate of change of the cows weight is 4kg a day. The market price is currently 50€ per kg and it is falling 50 cents a day. How much time should the cattleman wait to sell the cows? How much extra money does he make by waiting?

Aedan Tyler 2022-05-10 Answered

I have read a paper where, at some point, the following optimization problem appears:
$\begin{aligned} max_{x} w^{T} x \\ subject to {‖ x ‖}_{\infty} \leq ϵ \end{aligned}$
where $x$ and $w$ are real vectors and $ϵ$ is some given positive constant. The authors claim that the solution for this problem is $x = sign (w)$ , which seems clearly wrong to me, since then we would have ${‖ x ‖}_{\infty} = 1$ .

I believe the correct answer is $x = ϵ sign (w)$ . However, I am not being able to prove it. I have tried KKT multipliers, with no success. Any ideas?

poklanima5lqp3 2022-05-10 Answered

Consider that I have two items. Let $P_{i} (x_{i})$ denote the profit I get from item i by taking $x_{i}$ units of it (assume that $x_{i}$ can be a real number). Let $W_{i} (x_{i})$ denote the weight consumed by item i when I take $x_{i}$ units of this. Assume that both profit and weight functions are monotonically increasing in the quantity of items picked. The problem I am interested in is
$\begin{aligned} max_{x_{1}, x_{2}} & P_{1} (x_{1}) + P_{2} (x_{2}) \\ s . t . & W_{1} (x_{1}) + W_{2} (x_{2}) \leq W \end{aligned}$
where $W$ is total weight allowed. Thus I need to find $x_{i}$ (quantity of item $i$ ) such that the total profit is maximized while keeping the total weight under a given quantity. Let $x_{1}^{}$ and $x_{2}^{}$ be the solution to this problem. Is it true that
$\frac{P_{1} (x_{1}^{})}{W_{1} (x_{1}^{})} = \frac{P_{2} (x_{2}^{})}{W_{2} (x_{2}^{})}$
Note that ratio is essentially "Profit Per unit weight of an item". Assume that this were different at the optimum and the first item had the larger ratio. Thus I can get more profit per unit weight out of the first item. Then increasing quantity of that should increase profit. In order to balance the weights, I decrease the second items quantity decreasing its profit as well. But since the increase in profits from item 1 is higher, this should compensate for loss from item 2 and make the profits even more higher. What is flawed with this logic?

Hailee Stout 2022-05-10 Answered

The maximization problem is:
Maximize $u (x_{1}, x_{2}) = min [a_{1} x_{1}, a_{2} x_{2}] s.t. p_{1} x_{1} + p_{2} x_{2} \leq$ $w$ , in which $x_{i}, p_{i}$ is the amount and price of good $i$ , $w$ is the total budget available.
What I have been told to deal with this min[.,.] function is to solve it graphically. It's very easy to see on the graph that the maximization happens when $a_{1} x_{1} = a_{2} x_{2}$ . But I wonder if there is a way to solve this algebraically? I'm stumped at the first step, which is to derive $\frac{\partial u (x_{i})}{\partial x_{i}}$ .

kromo8hdcd 2022-05-09 Answered

For some strictly convex functions $f_{1} (x_{1}), f_{2} (x_{2}), . . ., f_{n} (x_{n})$ , it seems intuitive that for a given sum of $x_{1} + x_{2} + . . . + x_{n} = X$ , where $x_{1}$ , $x_{2}$ ,..., $x_{n} \geq 0$
$\exists k \in {1, 2, \dots, n}, f_{k} (X) = max {\sum_{i = 1}^{n} f_{i} (x_{i}) : \sum_{i = 1}^{n} x_{i} = X and \forall i, x_{i} \geq 0}$
Since these functions are convex, the sum of these functions is greatest when the sum of arguments is equal to the argument of just one function. This seems intuitive, but I can't think of how to argue this in formal maths. Any suggestions?

Laila Andrews 2022-05-09 Answered

How can I find $λ_{H}$ and $λ_{T}$ such that
$max_{0 \leq λ_{H}, λ_{T} \leq 1} {\frac{4.6575342 \times 10^{- 4}}{2.1722965 \times 10^{- 4} + λ_{H}}, \frac{1.0958904 \times 10^{- 2}}{3.4311896 \times 10^{- 4} + λ_{T}}} < 1 ?$
Is this problem equivalent to finding $x$ and $y$ such that
$min_{0 \leq x, y \leq 1} {.4664048 + (2147.0588450) x, 0.0313096 + (91.2500009) y} \geq 1 ?$

Oberhangaps5z 2022-05-09 Answered

A huge conical tank to be made from a circular piece of sheet metal of radius 10m by cutting out a sector with vertex angle theta and welding the straight edges of the straight edges of the remaining piece together. Find theta so that the resulting cone has the largest possible volume.

Specifically, the question is asked in the context of wanting derivatives, multiple max/min equations, and hopefully more calc rather than trig or geo.

I have gotten as far as using 10m as the hypotenuse for a triangle formed by the height of the cone, radius of the base of the cone, and slant. I'm not sure where to go from there, because I can't determine how to find height and/or radius, without which I'm not sure I can continue.

It will include more complex tasks to solve like congruence equation solving, which also becomes easier when you have a good example like what we provide.

High school geometry questions and answers