New SAT Math - Calculator - SAT Math

Card 0 of 1994

Question

Which of the following graphs correctly represents the quadratic inequality below (solutions to the inequalities are shaded in blue)?

$y\geq2(x+1)^2$

Answer

To begin, we analyze the equation given: the base equation, $y=x^{2}$ is shifted left one unit and vertically stretched by a factor of 2. The graph of the equation $y\geq2(x+1)^2$ is:

To solve the inequality, we need to take a test point and plug it in to see if it matches the inequality. The only points that cannot be used are those directly on our parabola, so let's use the origin . If plugging this point in makes the inequality true, then we shade the area containing that point (in this case, outside the parabola); if it makes the inequality untrue, then the opposite side is shaded (in this case, the inside of the parabola). Plugging the numbers in shows:

$0\geq2(0+1)^2$

Simplified as:

$0\geq2$

Which is not true, so the area inside of the parabola should be shaded, resulting in the following graph:

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Question

What is the average number of apples a student has?

Answer

To calculate the average number of apples a student has, the following formula is used.

$\text{Average } # \text{of Apples }=\frac{\text{Sum of Apples}}{\text{Sum of Students}}$

First, calculate the total number of apples there are. To do this multiply the number of apples by the number of students that have that many apples.

$\\text{Total Apples}=3(60)+2(30)+1(10) \\text{Total Apples}=180+60+10 \\text{Total Apples}=250$

This number divided by the total number of students.

$\text{Average } # \text{of Apples }=\frac{\text{250}}{60+30+10}=\frac{250}{100}=2.5$

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Question

The equation represents a line. This line does NOT pass through which of the four quadrants?

Answer

Plug in for to find a point on the line:

Thus, is a point on the line.

Plug in for to find a second point on the line:

is another point on the line.

Now we know that the line passes through the points and .

A quick sketch of the two points reveals that the line passes through all but the third quadrant.

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Question

Jesse has a large movie collection containing X movies. 1/3 of his movies are action movies, 3/5 of the remainder are comedies, and the rest are historical movies. How many historical movies does Jesse own?

Answer

1/3 of the movies are action movies. 3/5 of 2/3 of the movies are comedies, or (3/5)*(2/3), or 6/15. Combining the comedies and the action movies (1/3 or 5/15), we get 11/15 of the movies being either action or comedy. Thus, 4/15 of the movies remain and all of them have to be historical.

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Question

The length of an arc, , of a circle is $8\pi$ and the radius, , of the circle is . What is the measure in degrees of the central angle, $\theta$ , formed by the arc ?

Answer

The circumference of the circle is $2\pi r$ .

$2\pi(16)=32\pi$

The length of the arc S is $8\pi$ .

A ratio can be established:

$\frac{S}{\text{circumference}}=\frac{\theta}{360^o}$

$\frac{8\pi}{32\pi}=\frac{\theta}{360^o}$

Solving for _ $\theta$ _yields 90o.

Note: This makes sense. Since the arc S was one-fourth the circumference of the circle, the central angle formed by arc S should be one-fourth the total degrees of a circle.

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Question

Jessica wishes to fill up a cylinder with water at a rate of gallons per minute. The volume of the cylinder is gallons. The hole at the bottom of the cylinder leaks out gallons per minute. If there are gallons in the cylinder when Jessica starts filling it, how long does it take to fill?

Answer

Jessica needs to fill up gallons at the effective rate of $1.5:\text{gal/min}$ . $12: \text{gal}$ divided by $1.5: \text{gal/min}$ is equal to $8 :\text{min}$ . Notice how the units work out.

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Question

A vase needs to be filled with water. If the vase is a cylinder that is $\dpi{100} \small 12{}''$ tall with a $\dpi{100} \small 2{}''$ radius, how much water is needed to fill the vase?

Answer

Cylinder

$\dpi{100} \small V = \pi r^{2}h$

$\dpi{100} \small V = \pi (2)^{2}\times 12$

$\dpi{100} \small V = 4\times 12\pi$

$\dpi{100} \small V = 48\pi$

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Question

$\dpi{100} \small -2y+7>-7+y$

Given the inequality above, which of the following MUST be true?

Answer

$\dpi{100} \small -2y+7>-7+y$ Subtract from both sides:

Subtract 7 from both sides:

Divide both sides by $\dpi{100} \small -3$ :

$\frac{-3y}{-3}>\frac{-14}{-3}$

Remember to switch the inequality when dividing by a negative number:

$y<\frac{14}{3}$

Since $\dpi{100} \small y<\frac{14}{3}$ is not an answer, we must find an answer that, at the very least, does not contradict the fact that is less than (approximately) 4.67. Since any number that is less than 4.67 is also less than any number that is bigger than 4.67, we can be sure that is less than 5.

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Question

Find the roots of the following equation:

$3x^2+7x+2{}$

Answer

The equation that is given can be factor into:

The roots is the locations where this equation equals zero as seen below:

This occurs when the value in either parenthesis equals zero.

$(3x+1)=0 \and\ (x+2)=0$

Solving for the first expression:

$x=-\frac{1}{3}$

Solving for the second root:

Therefore the roots are:

$-\frac{1}{3},-2$

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Question

If $x+1< 4$ and $y-2<-1$ , then which of the following could be the value of ?

Answer

To solve this problem, add the two equations together:

$x+1<4$

$y-2<-1$

$x+1+y-2<4-1$

$x+y-1<3$

$x+y<4$

The only answer choice that satisfies this equation is 0, because 0 is less than 4.

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Question

$x^{2}-5x-24=0$

Solve for .

Answer

$x^{2}-5x-24=0$

Find all factors of 24

1, 2, 3,4, 6, 8, 12, 24

Now find two factors that add up to and multiply to ; and are the two factors.

By factoring, you can set the equation to be

If you FOIL it out, it gives you $x^{2}-5x-24=0$ .

Set each part of the equation equal to 0, and solve for .

and

and

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Question

A square $A$ has side lengths of $z$ . A second square $B$ has side lengths of $2.25z$ . How many $A's$ can you fit in a single $B$ ?

Answer

The area of $A$ is $n$ , the area of $B$ is $5.0625n$ . Therefore, you can fit 5.06 $A's$ in $B$ .

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Question

Solve for x

$\small 3x+7 \geq -2x+4$

Answer

$\small 3x+7 \geq -2x+4$

$\small 3x \geq -2x-3$

$\small 5x \geq -3$

$\small x\geq -\frac{3}{5}$

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Question

Sally sells custom picture frames. Her monthly fixed costs are $350. It costs $10 to make each frame. Sally sells her picture frames for $35 each.

How many picture frames must Sally sell in order to break even?

Answer

The break-even point is where the costs equal the revenues.

Let = # of frames sold

Costs:

Revenues:

Thus,

So 14 picture frames must be sold each month to break-even.

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Question

A family with 6 children, aged 4, 4, 5, 7, 12, and 13 are moving to a new home. They all want the same bedroom, so the parents have a lottery. Each child places their name in once for every year of age (the four year olds each put their name in 4 times, the seven year old puts his name in 7 times, etc.) What is the probability of the chosen child being 4 years old?

Answer

First, we will determine the total number of ballots:

$4+4+5+7+12+13=45\hspace{1 mm}ballots$

Since there are two four year olds, and this question is asking the probability of the chosen child being four, the probability is:

$\frac{4+4}{45}=\frac{8}{45}=0.1\overline{7}=17.\overline{7}%$

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Question

Jamie is three times her little brother's age, and her little brother is two years younger than his older brother. Collectively, the three of them are 27 years old. How old is Jamie?

Answer

The algebraic expression for being Jamie's youngest brother's age is:

Jamie's youngest brother is five, the next oldest brother is seven, and Jamie is 15.

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Question

How many negative solutions are there to the equation below?

Answer

First, subtract 3 from both sides in order to obtain an equation that equals 0:

The left side can be factored. We need factors of that add up to . and work:

Set both factors equal to 0 and solve:

$x - 1 = 0 \qquad \mbox{ or } \qquad x+3 =0$

To solve the left equation, add 1 to both sides. To solve the right equation, subtract 3 from both sides. This yields two solutions:

$x=1 \qquad x = -3$

Only one of these solutions is negative, so the answer is 1.

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Question

An amusement park charges both an entrance fee, and a fee for every ride. This fee is the same for all rides. Lisa went on 6 rides and paid 120 dollars. Tom went on only 4 rides and paid 95 dollars. What was the entrance fee?

Answer

We need 2 equations, because we have 2 unkown variables. Let = the entrance fee, and = the fee per ride. One ride costs dollars. We know that Lisa spent 120 dollars in total. Since Lisa went on 6 rides, she spent dollars on rides. Her only other expense was the entrance fee, :

Apply similar logic to Tom:

Subtracting the second equation from the first equation results in:

Divide both sides by 2:

So every ride costs 12.5 dollars. Plugging 12.5 back into one of the original equations allows us to solve for the entrance fee:

Subtract 50 from both sides:

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Question

Fred has $100 in quarters and nickels. He initially has 260 quarters. He then exchanges some of his nickels for the dimes of a friend. He is left with a total of 650 coins (consisting of quarters, dimes and nickels) still worth $100. How many nickels does Fred have now?

Answer

Fred has $100 in quarters and nickels initially. We are also told that he has 260 quarters. This is worth $65. Thus Fred initially has $35 in nickels or 700 nickels.

Fred now exchanges some of his nickels for the dimes of a friend. He ends up with 650 coins. We know that Fred started with 960 coins (700 nickels + 260 quarters). He ends up with 650 coins. The number of quarters remains unchanged, meaning he now has 390 nickels and dimes. These must have the same value as the initial 700 nickels, though, since he didn't lose any money.

Now we can finally set up our solution:

$35 = 39-.05x \Rightarrow .05x = 4 \Rightarrow x = 80$

Thus Fred has 80 nickels and 310 dimes.

$\line(1,0){250}$

An alternative solution step is to notice that turning nickels into dimes always occurs in exactly one way: 2 nickels to 1 dime. Every time you do this conversion, you will lose exactly one coin. We then notice that the number of coins drops from 960 to 650, or drops by 310 coins. We thus need to get rid of 310 coins. Since we're only allowed to change nickels into dimes (and lose 1 coin each time), we simply do this 310 times to reach the requisite number of coin losses. We are left with the proper number of coins with the proper value immediately. Since every replacement replaced 2 nickels, we also lost $310\cdot 2=620$ nickels. Our final number of nickels is thus nickels.

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Question

The graph below is the graph of a piece-wise function in some interval. Identify, in interval notation, the decreasing interval.

Answer

As is clear from the graph, in the interval between ( included) to , the is constant at and then from ( not included) to ( not included), the is a decreasing function.

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