Dr. Evers is experimenting with light beams and prisms. He passes a beam of white light through a triangular prism which spreads the light out into its six rainbow colors. The bases of the prism are equilateral triangles. The surface area of this prism is 4,292 square millimeters. The area of each triangular face is 271 square millimeters. Which expression can be used to find h, the height, in millimeters, of the prism?
- A. 4,292/3(25)
- B. 4,292/271
- C. (4,292-271)/25
- D. (4,292-2(271))/3(25)
Correct Answer & Rationale
Correct Answer: D
To find the height \( h \) of the prism, we start with the total surface area of the prism, which includes the two triangular bases and three rectangular sides. The area of the two triangular bases is \( 2 \times 271 = 542 \) square millimeters. Subtracting this from the total surface area gives \( 4,292 - 542 = 3,750 \) square millimeters for the area of the rectangular sides. Since the height \( h \) is involved in the area of the rectangles, dividing this area by the perimeter of the base (which is \( 3 \times 25 = 75 \) mm) leads to \( h = \frac{3,750}{75} \) or \( \frac{4,292 - 542}{75} \), simplifying to option D. Options A and B incorrectly compute the height without accounting for the rectangular areas properly. Option C miscalculates the area of the triangular bases and does not consider the full surface area needed to find \( h \). Thus, only option D correctly utilizes the total surface area and the dimensions of the prism to derive the height.
To find the height \( h \) of the prism, we start with the total surface area of the prism, which includes the two triangular bases and three rectangular sides. The area of the two triangular bases is \( 2 \times 271 = 542 \) square millimeters. Subtracting this from the total surface area gives \( 4,292 - 542 = 3,750 \) square millimeters for the area of the rectangular sides. Since the height \( h \) is involved in the area of the rectangles, dividing this area by the perimeter of the base (which is \( 3 \times 25 = 75 \) mm) leads to \( h = \frac{3,750}{75} \) or \( \frac{4,292 - 542}{75} \), simplifying to option D. Options A and B incorrectly compute the height without accounting for the rectangular areas properly. Option C miscalculates the area of the triangular bases and does not consider the full surface area needed to find \( h \). Thus, only option D correctly utilizes the total surface area and the dimensions of the prism to derive the height.
Other Related Questions
John and Mike are participating in a long-distance bicycling event. Mike bicycled 24 miles in the first 2 hours. The distance John has bicycled over the first 11 minutes is shown in the chart. If John and Mike continue at the same rates, which statement will be true about their distances 4 hours into the event?
- A. John will be 6 miles ahead of Mike.
- B. John will be 12 miles ahead of Mike.
- C. Mike will be 6 miles ahead of John.
- D. Mike will be 12 miles ahead of John.
Correct Answer & Rationale
Correct Answer: D
To determine who is ahead after 4 hours, we first calculate the speeds of both cyclists. Mike's speed is 12 miles per hour (24 miles in 2 hours). In 4 hours, he will cover 48 miles (12 mph x 4 hours). John's distance after 11 minutes (or 0.183 hours) needs to be extrapolated. If he biked 3 miles in that time, his speed is approximately 16 miles per hour (3 miles ÷ 0.183 hours). Over 4 hours, John would cover about 64 miles (16 mph x 4 hours). Comparing their distances: John at 64 miles and Mike at 48 miles means Mike is 12 miles behind John, confirming option D is accurate. Options A and B incorrectly suggest John is ahead, while C miscalculates Mike's lead.
To determine who is ahead after 4 hours, we first calculate the speeds of both cyclists. Mike's speed is 12 miles per hour (24 miles in 2 hours). In 4 hours, he will cover 48 miles (12 mph x 4 hours). John's distance after 11 minutes (or 0.183 hours) needs to be extrapolated. If he biked 3 miles in that time, his speed is approximately 16 miles per hour (3 miles ÷ 0.183 hours). Over 4 hours, John would cover about 64 miles (16 mph x 4 hours). Comparing their distances: John at 64 miles and Mike at 48 miles means Mike is 12 miles behind John, confirming option D is accurate. Options A and B incorrectly suggest John is ahead, while C miscalculates Mike's lead.
Last weekend, 625 runners entered a 10,000-meter race. A 10,000- meter race is 6.2 miles long. Ruben won the race with a finishing time of 29 minutes 51 seconds.
The graphs show information about the top 10 runners.
Based on the scatter plot, what is the range of ages of the top 10 runners?
- A. 9
- B. 1
- C. 16
- D. 40
Correct Answer & Rationale
Correct Answer: C
The range of ages is determined by subtracting the youngest runner's age from the oldest runner's age. In this case, the scatter plot indicates that the youngest runner is 16 years old and the oldest is 32 years old. Thus, the range is 32 - 16 = 16 years. Option A (9) incorrectly suggests a smaller age difference, while B (1) implies almost no age variation, neither of which aligns with the data presented. Option D (40) overestimates the age range, indicating a misunderstanding of the plotted values. Therefore, the accurate calculation of 16 years reflects the true age span of the top 10 runners.
The range of ages is determined by subtracting the youngest runner's age from the oldest runner's age. In this case, the scatter plot indicates that the youngest runner is 16 years old and the oldest is 32 years old. Thus, the range is 32 - 16 = 16 years. Option A (9) incorrectly suggests a smaller age difference, while B (1) implies almost no age variation, neither of which aligns with the data presented. Option D (40) overestimates the age range, indicating a misunderstanding of the plotted values. Therefore, the accurate calculation of 16 years reflects the true age span of the top 10 runners.
Simplify: (3x - 5) + (-7x + 2)
- A. -4x^2 - 3
- B. -4x - 3
- C. 28
- D. -4x^2 - 10
Correct Answer & Rationale
Correct Answer: B
To simplify the expression (3x - 5) + (-7x + 2), first combine like terms. Start with the x terms: 3x + (-7x) results in -4x. Next, combine the constant terms: -5 + 2 equals -3. Thus, the simplified expression is -4x - 3, matching option B. Option A, -4x^2 - 3, incorrectly includes an x^2 term that does not exist in the original expression. Option C, 28, is unrelated to the simplification process. Option D, -4x^2 - 10, also includes an incorrect x^2 term and miscalculates the constants.
To simplify the expression (3x - 5) + (-7x + 2), first combine like terms. Start with the x terms: 3x + (-7x) results in -4x. Next, combine the constant terms: -5 + 2 equals -3. Thus, the simplified expression is -4x - 3, matching option B. Option A, -4x^2 - 3, incorrectly includes an x^2 term that does not exist in the original expression. Option C, 28, is unrelated to the simplification process. Option D, -4x^2 - 10, also includes an incorrect x^2 term and miscalculates the constants.
Tina Is designing a cabin. One of her plans for the cabin is a rectangle twice as long as it is wide, with 10 feet (ft) of the length reserved for the Kitchen and the bathroom. The diagram shows this basic plan. Tina wants the area of the main room to be 300 square feet. Which equation can be used to find x, the width, in feet, of the main room?
- A. 2x^2 + 10x - 300 = 0
- B. 2x^2 - 10x - 300 = 0
- C. 2x^2 - 20x - 300 = 0
- D. 2x^2 + 20x - 300 = 0
Correct Answer & Rationale
Correct Answer: B
To determine the width \( x \) of the main room, we start with the area formula for a rectangle: Area = Length × Width. The cabin's length is twice the width, so it can be expressed as \( 2x \). Since 10 ft is allocated for the kitchen and bathroom, the length of the main room is \( 2x - 10 \). The equation for the area of the main room is therefore \( (2x - 10)x = 300 \), which simplifies to \( 2x^2 - 10x - 300 = 0 \), matching option B. Option A incorrectly adds \( 10x \) instead of subtracting, leading to an incorrect area calculation. Option C miscalculates the length by subtracting 20 instead of 10, while option D incorrectly adds 20, which does not reflect the reserved space. Thus, only option B accurately represents the relationship between length, width, and area.
To determine the width \( x \) of the main room, we start with the area formula for a rectangle: Area = Length × Width. The cabin's length is twice the width, so it can be expressed as \( 2x \). Since 10 ft is allocated for the kitchen and bathroom, the length of the main room is \( 2x - 10 \). The equation for the area of the main room is therefore \( (2x - 10)x = 300 \), which simplifies to \( 2x^2 - 10x - 300 = 0 \), matching option B. Option A incorrectly adds \( 10x \) instead of subtracting, leading to an incorrect area calculation. Option C miscalculates the length by subtracting 20 instead of 10, while option D incorrectly adds 20, which does not reflect the reserved space. Thus, only option B accurately represents the relationship between length, width, and area.