Answer:
47.88 in^2 of paper.
Step-by-step explanation:
The amount of paper needed for the base = 4.2^2 = 17.64 in^2.
There are 4 triangular faces.
The area of each triangular face = 1/2* base * height
= 1/2 * 4.2 * 3.6
= 7.56 in^2
That is a total of 4 * 7.56 = 30.24 in^2.
So the total amount of paper need to cover the entire pyramid
= 17,64 + 30.24
= 47.88 in^2.
It took the race car 22 minutes to travel 114.4 kilometers. At what rate did the car travel? Use the formula r=d/t where r is the rate, d is the distance, and t is the time. Round your answer to the nearest tenth.
Answer:
5.2 Km/Min
Step-by-step explanation:
Speed/Rate= Distance/Time
Speed/Rate = 114.4/22
Speed/Rate = 5.2 KM/min
Answer:
5.3 km/min
Step-by-step explanation:
Distance = 114.4 kilometers
Time = 22 minutes
Rate = 114.4 km /22 min
Rate = 5.2 km/min
What is the area of this triangle?
Enter your answer in the box.
Answer:
A = 14 units ^2
Step-by-step explanation:
The area of a triangle is
A = 1/2 b h where b is the base and h is the height
b = the length of EF which is 7 units
h = D to the line EF which is 4 units
A = 1/2 (7*4)
A = 14 units ^2
The magnitude of the resultant vector shown is _____.
Answer:
2√3
Step-by-step explanation:
You recognize this as a 30°-60°-90° triangle, so you know the hypotenuse (R) is twice the length of the shortest side (√3).
The magnitude of R is 2√3.
_____
In case you haven't memorized the ratios for a 30°-60°-90° triangle, you can use trigonometry and the fact that ...
Sin = Opposite/Hypotenuse
sin(30°) = √3/R
R = √3/sin(30°) = √3/(1/2) = 2√3
Of course, doing this on your calculator will give a numerical answer, which you may not want.
A photo originally measuring 11 inches by 9 inches needs to be enlarged to a size of 55 by 45 inches. Find the scale factor.
Answer:
5
Step-by-step explanation:
Given
Original Measurement = 11*9 inches
Measurement after enlargement = 55 * 45 inches
In order to find the scale factor, we can choose one side of the figure or the whole area and find the ratio between the measurement before enlargement and after enlargement.
In case of a side the answer will be the scale factor while in case of finding scale factor using areas the answer will be the square of scale factor.
So,
[tex]Scale\ factor =s^2= \frac{55*45}{11*9} \\s^2 = \frac{2475}{99} \\s^2=25[/tex]
As we know that this is the square of scale factor.
Hence the scale factor will be:
[tex]\sqrt{s^2}=\sqrt{25} \\s=5[/tex]
So, the scale factor is 5 ..
Explain the difference between qualitative and quantitative data. Choose the correct answer below. A. Quantitative data are collected from a designed experiment, while qualitative data are from an observational study. B. Quantitative data are collected from an observational study, while qualitative data are from a designed experiment. C. Quantitative data are categorical in nature, while qualitative data are numerical in nature. D. Quantitative data are data from a population, while qualitative data are data from a sample. E. Quantitative data are data from a sample, while qualitative data are data from a population. F. Quantitative data are numerical in nature, while qualitative data are categorical in nature.
The correct answer is F. Quantitative data are numerical in nature, while qualitative data are categorical in nature.
Explanation:
In research and all the different fields that apply to it, the word "data" refers to information, values or knowledge that can be used to understand a specific situation or phenomenon. Additionally, data can be of two different types quantitative and qualitative, these differ in their nature, the phenomenons they described and the way they should be analyzed. Indeed quantitative data refers mainly to numerical data or information about quantities such as statistics that are especially useful in mathematics, science and similar that focus on numbers. On the other hand, qualitative data refers to data based on categories or qualities and because of this qualitative data is used in humanistic research, although both types of data can be combined to study a phenomenon. Considering this, the key difference between both types of data is "Quantitative data are numerical in nature, while qualitative data are categorical in nature".
Answer:
it is F
Step-by-step explanation:
F. Quantitative data are numerical in nature, while qualitative data are categorical in nature.
At what points on the given curve x = 4t3, y = 3 + 8t − 10t2 does the tangent line have slope 1?
Answer:
(4/3, 4 5/9) and (-32, -53)
Step-by-step explanation:
When a curve is given as a set of parametric equations, as this one is, then the slope of the tangent line to the curve is
dy/dt
dy/dx = ------------
dx/dt
which here is
dy/dt 8 - 20t
dy/dx = ----------- = --------------
dx/dt 12t^2
If the slope at a certain point on this curve is 1, then we conclude that:
8 - 20t = 12t^2, or
12t^2 + 20t - 8 = 0, or
3t^2 + 5t - 2 = 0
We have to solve this equation for the parameter, t:
Here a = 3, b = 5 and c = -2, and so the discriminant is
b^2 - 4ac = 25 - 4(3)(-2), or 49, and the square root of that is 7.
Thus, the roots are:
-5 ± 7
t = --------- = 1/3 and t = -2
2(3)
Evaluate x and y twice, once each for each t value.
Case 1: t = 1/3
x = 4(1/3) and y = 3 + 8(1/3) - 10(1/3)^2, or
x = 4/3 and y = 3 + 8/3 - 10/9: (4/3, 4 5/9)
Case 2: t = -2
x = 4(-2)^3 and y = 3 + 8(-2) - 10(-2)^2, or y = 3 - 16 - 40, or y = -53.
This gives us the point (-32, -53)
To find the points where the tangent line has a slope of 1 on the given curve, we derive expressions for dx/dt and dy/dt, set dy/dx equal to 1, and solve for t.
Explanation:The given equation defines a parametric curve with x = 4t3 and y = 3 + 8t − 10t2. To find the points on this curve where the tangent line has a slope of 1, we'll first need to find expressions for dx/dt and dy/dt, which represent the rates of change of x and y with respect to the parameter t.
By differentiating x = 4t3 with respect to t, we find dx/dt = 12t2. Similarly, by differentiating y = 3 + 8t − 10t2 with respect to t, we find dy/dt = 8 - 20t.
The slope of the tangent line at a particular point on the curve corresponds to dy/dx, which we can find by dividing dy/dt by dx/dt, yielding (8 - 20t) / (12t2). We can set this equal to 1 (since we want a slope of 1) and solve for t to find the points on the curve where the tangent line has slope 1.
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Find the value of b.
Answer:
b ≈ 17
Step-by-step explanation:
Using trigonometric ratio, sine we can find side b.
sine x = opposite/hypotenuse
sin 45 = 12/b
b ≈ 17
Answer:
b = 12[tex]\sqrt{2}[/tex]
Step-by-step explanation:
Using the sine ratio in the right triangle and the exact value
sin45° = [tex]\frac{1}{\sqrt{2} }[/tex]
From the triangle
sin45° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{12}{b}[/tex] = [tex]\frac{1}{\sqrt{2} }[/tex]
Cross- multiply
b = 12[tex]\sqrt{2}[/tex]
If you want to prove that the diagonals of a parallelogram bisect each other using coordinate geometry, how would you place the parallelogram on the coordinate plane? Give the coordinates of the vertices for the placement you choose.
Answer:
In general you can choose the vertices at any arbitrary points but for easier computations and calculations we can choose 1 vertex at origin with co-ordinates [tex](0,0)[/tex] and it's adjacent vertex either on x-axis with co-ordinates [tex](x,0)[/tex] or on y-axis with ordinates [tex](0,y)[/tex]
Thus the coordinates of vertices become
Brianna went to a carnival. She played five games and rode six rides. Justin went to the same carnival and played seven games and rode eight rides. If Brianna paid $11.75 and Justin paid $16.15, how much did one game cost to play?
$0.50
$1.45
$1.25
$1.50
Urgent!
Answer:
$1.45
Step-by-step explanation:
To answer the question, we would like to have an equation that has the cost of a game as its only variable. That is, we would like to eliminate the cost of a ride from the system of equations we must write.
Let g and r represents the costs of a game and a ride, respectively. Then the expenses of the two carnival-goers can be described by ...
5g +6r = 11.75
7g +8r = 16.15
We note that the ratio of coefficients in the variable (r) that we want to eliminate is 3:4. So we can subtract 4 times the first equation from 3 times the second to eliminate that variable.
3(7g +8r) -4(5g +6r) = 3(16.15) -4(11.75)
g = 1.45 . . . . . . simplify
The cost to play one game was $1.45.
Answer:
$1.45
Step-by-step explanation:
Darcie wants to crochet a minimum of 3 blankets to donate to a homeless shelter. Darcie crochets at a rate of 1/15, end of a blanket per day. She has 60 days until when she wants to donate the blankets, but she also wants to skip crocheting some days so she can volunteer in other ways.Write an inequality to determine the number of days, Darcie can skip crocheting and still meet her goal.
Answer:
x≤ 60 - (3÷1/15)
Step-by-step explanation:
Number of blankets to be made = 3
Darcie crochets 1/15 blanket per day
To crochet one carpet she needs = 15 days
To crochet three carpets she needs = 15*3 = 45 days
Number of days she had = 60 days
She can skip days = 60-45 = 15 days
Let x be the number of days to complete her work.
Thus the equation becomes x≤ 60 - (3÷1/15)
You can solve for x to determine the number of days, Darcie can skip crocheting, you will get the answer 15....
The answer explains how to set up and solve an inequality to determine the number of days Darcie can skip crocheting while still meeting her goal of donating blankets.
To determine the number of days Darcie can skip crocheting and still meet her goal, we need to set up an inequality based on the information given.
First, calculate the total number of blankets Darcie needs to crochet: 3 blankets.
Set up the inequality: 1/15 * (60 - x) ≥ 3, where x represents the number of days she can skip crocheting.
Solve the inequality: 60 - x ≥ 45, x ≤ 15.
Therefore, Darcie can skip crocheting for up to 15 days and still meet her goal of donating 3 blankets.
Which is a zero of the quadratic function f(x) = 9x2 – 54x – 19?
x =
x = 3
x = 6
x = 9
Answer: x=-3
Step-by-step explanation:
F(x)=9x^2-54x-19
F(x)=(x-57)(x+3)
X=57 X=-3
If AD = 26 and AB = 24, calculate length of line segment BD. Segment AC is tangent to circle D.
Answer:
BD = 10
Step-by-step explanation:
Since AC is a tangent to the circle at B then ∠ABD = 90°
Using Pythagoras' identity in the right triangle ABD with hypotenuse AD
The square on the hypotenuse is equal to the sum of the squares on the other two sides, that is
BD² + AB² = AD²
BD² + 24² = 26²
BD² + 576 = 676 ( subtract 576 from both sides )
BD² = 100 ( take the square root of both sides )
BD = [tex]\sqrt{100}[/tex] = 10
Properties of shape. Need help on this question!!
Answer:
The right answer us "is always" as its all angles are 60 degree.
Examine the system of equations. 2x + y = 34 -3x + 1 2 y = 25 If you multiply the first equation by 2, what must you multiply the second equation by to eliminate the y-variable.
answer: -4
-2
1
4
Answer:
Lol you answered your own question but yes it is -4 thanks!
Step-by-step explanation:
What is the difference of the polynomials?
(–2x3y2 + 4x2y3 – 3xy4) – (6x4y – 5x2y3 – y5)
Answer:
[tex]\large\boxed{-2x^3y^2+9x^2y^3-3xy^4-6x^4y+y^5}[/tex]
Step-by-step explanation:
[tex](-2x^3y^2+4x^2y^3-3xy^4)-(6x^4y-5x^2y^3-y^5)\\\\=-2x^3y^2+4x^2y^3-3xy^4-6x^4y+5x^2y^3+y^5\qquad\text{combine like terms}\\\\=-2x^3y^2+\underline{4x^2y^3}-3xy^4-6x^4y+\underline{5x^2y^3}+y^5\\\\=-2x^3y^2+9x^2y^3-3xy^4-6x^4y+y^5[/tex]
Suppose we want to divide the 10 dogs into three groups, one with 3 dogs, one with 5 dogs, and one with 2 dogs. how many ways can we form the groups such that fluffy is in the 3-dog group and nipper is in the 5-dog group?
Answer:
420 ways
Step-by-step explanation:
According to the given statement:
We want to divide the 10 dogs into three groups, one with 3 dogs, one with 5 dogs, and one with 2 dogs. How many ways can we form the groups such that fluffy is in the 3-dog group and nipper is in the 5-dog group.
In this way we have 8 dogs left.
2 spaces left in 3 dogs group
4 spaces left in 5 dogs group
and 2 spaces in 2 dogs group
Therefore:
= 8!/2!4!2!
= 8*7*6*5*4*3*2*1/2*4*3*2*2
= 8*7*6*5/2*2
= 1680/4
=420
It means there are 420 ways to from the groups....
A catering service offers
6
appetizers,
7
main courses, and
10
desserts. A customer is to select
5
appetizers,
4
main courses, and
5
desserts for a banquet. In how many ways can this be done?
There are 52,920 ways to select 5 appetizers, 4 main courses, and 5 desserts for the banquet.
To solve this problem, we can use the concept of combinations, as we're selecting items without considering the order.
For appetizers:
We need to choose 5 appetizers out of 6 available. This can be calculated using the combination formula: nCr = n! / [r! * (n-r)!], where n is the total number of items, and r is the number of items to be chosen.
So, for appetizers, it's [tex]6C_5 = 6! / [5! * (6-5)!] = 6 ways[/tex].
For main courses:
Similarly, we need to choose 4 main courses out of 7 available. So, it's
[tex]7C_4 = 7! / [4! * (7-4)!] = 35 ways[/tex].
For desserts:
We need to choose 5 desserts out of 10 available. So, it's
[tex]10C_5 = 10! / [5! * (10-5)!] = 252 ways[/tex].
To find the total number of ways:
We multiply the number of ways for each category since these choices are independent.
Total ways = (6 ways for appetizers) * (35 ways for main courses) * (252 ways for desserts) = 52920 ways.
Thus, there are 52,920 ways to select 5 appetizers, 4 main courses, and 5 desserts for the banquet.
Complete Question:
A catering service offers 6 appetizers, 7 main courses, and 10 desserts. A customer is to select 5 appetizers, 4 main courses, and 5 desserts for a banquet. In how many ways can this be done?
Answer if you can :)
If f(x) = -7x – 3 and g(x) = radical over x+6,
what is (fºg)(-2)
Answer:
-17
Step-by-step explanation:
Plug in -2 as your x value for the g(x) equation and simplify.
[tex]g(-2)=\sqrt{-2+6} \\g(-2)=\sqrt{4} \\g(-2)=2[/tex]
Next, plug in your g(x) value (2) to the f(x) equation for x and simplify.
[tex]f(2)=-7(2)-3\\f(2)=-14-3\\f(2)=-17[/tex]
tan(x - 3 π ) = _____
-1
1
-tanx
tanx
[tex]\bf \textit{Sum and Difference Identities} \\\\ tan(\alpha + \beta) = \cfrac{tan(\alpha)+ tan(\beta)}{1- tan(\alpha)tan(\beta)} \qquad tan(\alpha - \beta) = \cfrac{tan(\alpha)- tan(\beta)}{1+ tan(\alpha)tan(\beta)} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ tan(x-3\pi )=\cfrac{tan(x)-tan(3\pi )}{1+tan(x)tan(3\pi )} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf tan(3\pi )\implies \cfrac{sin(3\pi )}{cos(3\pi )}\implies \cfrac{0}{-1}\implies 0\qquad therefore \\\\[-0.35em] ~\dotfill\\\\ tan(3\pi )=\cfrac{tan(x)-tan(3\pi )}{1+tan(x)tan(3\pi )}\implies tan(x-3\pi )=\cfrac{tan(x)-0}{1+0} \\\\\\ tan(x-3\pi )=\cfrac{tan(x)}{1}\implies tan(x-3\pi )=tan(x)[/tex]
A line has a slope of -3/5.which order pairs could be points on a Parnell line
Answer:
yes
Step-by-step explanation:
this is right i think
The correct answer is line HJ .
1. **Line AB**: The slope of line AB is not the negative reciprocal of 1/2, so it is not perpendicular.
2. **Line CD**: The slope of line CD is not the negative reciprocal of 1/2, so it is not perpendicular.
3. **Line FG**: The slope of line FG is not the negative reciprocal of 1/2, so it is not perpendicular.
4. **Line HJ**: The slope of line HJ is the negative reciprocal of 1/2, which makes it perpendicular.
A line with a slope of **-3/5** can be represented by the equation:
[tex]\[ y = -\frac{3}{5}x + b \][/tex]
where \(b\) is the y-intercept. To find points on this line, we need to consider different values of \(x\) and calculate the corresponding \(y\).
Let's explore some potential points:
1. **Point A (x, y)**:
- Assume \(x = 0\):
[tex]\[ y = -\frac{3}{5} \cdot 0 + b = b \] - So, point A is \((0, b)\).[/tex]
2. **Point B (x, y)**:
- Assume \(x = 5\):
[tex]\[ y = -\frac{3}{5} \cdot 5 + b = -3 + b \] - So, point B is \((5, -3 + b)\).[/tex]
3. **Point C (x, y)**:
- Assume \(x = 10\):
[tex]\[ y = -\frac{3}{5} \cdot 10 + b = -6 + b \] - So, point C is \((10, -6 + b)\).[/tex]
These are just a few examples. You can find more points by choosing different values of \(x\). Remember that any point on the line will satisfy the equation [tex]\(y = -\frac{3}{5}x + b\).[/tex]
Now, let's explore the concept of parallel lines. Two lines are parallel if they have the **same slope**. If we have another line with a slope of 1/2, we can find points on that line as well.
Find the missing lengths in right triangle mno. Estimate your answer your answer to two decimal places..
Answer:
x = 11.79
? = 6.25
Step-by-step explanation:
Using the law of sines
[tex]\frac{10}{sin58}[/tex] = [tex]\frac{x}{sin90}[/tex]
x = sin 90° x [tex]\frac{10}{sin58}[/tex] = 11.79
by Pythagorean theorem,
?² + 10² = x²
? = √ (x²- 10²)
? = √ (11.79²- 10²) = 6.25
Lisa's penny bank is 1/10 full. After she adds 440 pennies, it is 3/5 full. How many pennies can Lisa's bank hold?
First lets change the fractions so they have common denominators:
3/5 = 6/10
The bank was 1/10, after adding the pennies it was 6/10
6/10 - 1/10 = 5/10 = 1/2
This means 400 pennies filled 1/2 the piggy bank.
There are 2 halves (1/2) to a whole ( full piggy bank).
The piggy bank can hold 400 x 2 = 800 pennies.
For the following system, if you isolated x in the second equation to use the substitution method, what expression would you substitute into the first equation?
3x + y = 8
−x − 2y = −10
A.) −2y + 10
B.) 2y + 10
C.) 2y − 10
D.) −2y − 10
Answer:
A.) −2y + 10
Step-by-step explanation:
−x − 2y = −10
Add 2y to both sides.
-x = 2y - 10
Multiply both sides by -1.
x = -2y + 10
Let's solve the system using the substitution method. We'll start by isolating \( x \) in the second equation.
The second equation is given by:
\[ -x - 2y = -10 \]
To isolate \( x \), we'll follow these steps:
1. Add \( 2y \) to both sides of the equation, which gives us:
\[ -x = 2y - 10 \]
2. Now, we multiply both sides by \( -1 \) to solve for \( x \), which gives us:
\[ x = -1(-2y + 10) \]
\[ x = 2y - 10 \]
The expression that we would substitute into the first equation for \( x \) is \( 2y - 10 \).
Therefore, the correct answer is:
C.) \( 2y - 10 \)
Mason opened a new electronic store, and his daily sales are modeled by f(x) = 50(1.2)x. Determine the rate of growth.
A.) 50%
B.) 20%
C.) 12%
D.) 120%
An exponential function may be given by:
f(x) = A(1+r)^x
A is the initial amount and r is a decimal representing the growth rate.
We can see that 1+r = 1.2, and we solve for r:
1 + r = 1.2
r = 0.2
The growth rate is 0.2, or 20%
Choice B
The rate of growth of Mason's daily sales is 20%.
Explanation:The rate of growth of Mason's daily sales can be found by determining the percentage increase in the sales from one day to the next. To find this, we can compare the sales on two consecutive days and calculate the ratio of the second day's sales to the first day's sales. Let's consider the sales on the first day (x) and the sales on the second day (x+1).
Given the sales model f(x) = 50(1.2)^x, we can substitute x and x+1 into the equation to get the sales on the first and second day, respectively. The ratio of the second day's sales to the first day's sales is:
f(x+1) / f(x) = (50(1.2)^(x+1)) / (50(1.2)^x) = 1.2.
So, the rate of growth is 1.2, which represents a 20% increase in sales from one day to the next.
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Find the value of x in the triangle shown below.
Answer:
37°
Step-by-step explanation:
By definition all internal angles of a triangle add up to 180°
Hence,
98° + 45° + x = 180°
x = 180° - 98° - 45° = 37°
Drag a number into each line to create an equation that is true for all values of x
Answer:
blank 1 = 24
blank2= 30
Step-by-step explanation:
Given:
2(4x+3)(3x+5)
=2(12x^2+20x+9x+15)
=2(12^2+29x+15)
=24x^2+58x+30
Hence blank 1= 24 and blank2= 30 !
Can someone please check to make sure I got this correct? I would appreciate if you showed your work so that I could compare with my work. Thank you!
Answer:
-4
Step-by-step explanation:
[√2(cos(3π/4) + i sin(3π/4))]⁴
(√2)⁴ (cos(3π/4) + i sin(3π/4))⁴
4 (cos(3π/4) + i sin(3π/4))⁴
Using De Moivre's Theorem:
4 (cos(4 × 3π/4) + i sin(4 × 3π/4))
4 (cos(3π) + i sin(3π))
3π on the unit circle is the same as π:
4 (cos(π) + i sin(π))
4 (-1 + i (0))
-4
Right cone A and oblique cone B both have a height of 26 mm. Complete the statements about the two cones. The volumes of the cones are equal when . If a > b, then the cross-sectional area of cone A is the cross-sectional area of cone B at every level parallel to their respective bases.
Answer:
a=b
greater than
Step-by-step explanation:
Answer:
volumes of the cones are equal when (a=b)
If a > b, then the cross-sectional area of cone a is (GREATER THAN) is the cross-sectional area of cone B at every level parallel to their respective bases.
Step-by-step explanation:
The box plots show the weights, in pounds, of the dogs in two different animal shelters.
Weights of Dogs in Shelter A (Top Plot)
Weights of Dogs in Shelter B(Bottom Plot)
Which correctly compares the ranges of the data?
The range in shelter A is 11, and the range in shelter B is 4.
The range in shelter A is 20, and the range in shelter B is 10.
The range in shelter A is 13, and the range in shelter B is 8.
The range in shelter A is 22, and the range in shelter B is 18.
Answer:
The range in shelter A is 22, and the range in shelter B is 18.
Step-by-step explanation:
The range is the largest value minus the smallest value.
For shelter A, the range is 30 − 8 = 22.
For shelter B, the range is 28 − 10 = 18.
Answer:
The range in shelter A is 22, and the range in shelter B is 18.
Step-by-step explanation:
The box plots show the weights, in pounds, of the dogs in two different animal shelters. The range in shelter A is 22, and the range in shelter B is 18 correctly compares the ranges of the data.
Is her assertion correct ?
Check the picture below.
so, the vertex at N, is noticeably not a right angle is an acute angle, so is less than 90°, so we don't need to check that one.
now, is the angle at L 90°?
well, if that's true LM and LN are perpendicular, and if they're indeed perpendicular, their slopes are negative reciprocal, meaning the slope of one is the same as the other but negative and upside down, well, let's check.
[tex]\bf L(\stackrel{x_1}{0}~,~\stackrel{y_1}{0})\qquad M(\stackrel{x_2}{2}~,~\stackrel{y_2}{2}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{2-0}{2-0}\implies \cfrac{2}{2}\implies 1 \\\\[-0.35em] ~\dotfill\\\\ L(\stackrel{x_1}{0}~,~\stackrel{y_1}{0})\qquad N(\stackrel{x_2}{2}~,~\stackrel{y_2}{-1}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{-1-0}{2-0}\implies \cfrac{-1}{2}\implies -\cfrac{1}{2}[/tex]
[tex]\bf \stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {\stackrel{slope~of~LM}{1\implies \cfrac{1}{\underline{1}}}\qquad \qquad \qquad \stackrel{reciprocal}{\cfrac{\underline{1}}{1}}\qquad \stackrel{negative~reciprocal}{-\cfrac{\underline{1}}{1}\implies -1}} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{slope of LM}}{1}\qquad \stackrel{\textit{negative reciprocal of LM}}{-1}\qquad \stackrel{\textit{slope of LN}}{-\cfrac{1}{2}}~\hfill -1\ne -\cfrac{1}{2}[/tex]
so that means Lydia put too much espresso on her last cup.